Probability RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Q u e s t i o n : 1
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
S o l u t i o n :
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 2
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
S o l u t i o n :
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is
115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Page 2


Q u e s t i o n : 1
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
S o l u t i o n :
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 2
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
S o l u t i o n :
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is
115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 3
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
Outcome: No head One head Two heads Three heads
Frequency: 14 38 36 12
If the three coins are simultaneously tossed again, compute the probability of:
i
2 heads coming up.
ii
3 heads coming up.
iii
at least one head coming up.
iv
getting more heads than tails.
v
getting more tails than heads.
S o l u t i o n :
The total number of trials is 100.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of getting two heads.
The number of times A happens is 36.
Therefore, we have
ii
Let B be the event of getting three heads
The number of times B happens is 12.
Therefore, we have
iii
Let C be the event of getting at least one head.
The number of times C happens is .
Page 3


Q u e s t i o n : 1
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
S o l u t i o n :
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 2
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
S o l u t i o n :
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is
115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 3
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
Outcome: No head One head Two heads Three heads
Frequency: 14 38 36 12
If the three coins are simultaneously tossed again, compute the probability of:
i
2 heads coming up.
ii
3 heads coming up.
iii
at least one head coming up.
iv
getting more heads than tails.
v
getting more tails than heads.
S o l u t i o n :
The total number of trials is 100.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of getting two heads.
The number of times A happens is 36.
Therefore, we have
ii
Let B be the event of getting three heads
The number of times B happens is 12.
Therefore, we have
iii
Let C be the event of getting at least one head.
The number of times C happens is .
Therefore, we have
iv
Let D be the event of getting more heads than tails.
The number of times D happens is .
Therefore, we have
v
Let E be the event of getting more tails than heads.
The number of times E happens is .
Therefore, we have
Q u e s t i o n : 4
1500 families with 2 children were selected randomly and the following data were recorded:
 
Number of girls in a family 0 1 2
Number of families 211 814 475
If a family is chosen at random, compute the probability that it has:
i
No girl
ii
1 girl
iii
2 girls
iv
at most one girl
v
more girls than boys
S o l u t i o n :
The total number of trials is 1500.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of having no girl.
The number of times A happens is 211.
Therefore, we have
Page 4


Q u e s t i o n : 1
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
S o l u t i o n :
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 2
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
S o l u t i o n :
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is
115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 3
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
Outcome: No head One head Two heads Three heads
Frequency: 14 38 36 12
If the three coins are simultaneously tossed again, compute the probability of:
i
2 heads coming up.
ii
3 heads coming up.
iii
at least one head coming up.
iv
getting more heads than tails.
v
getting more tails than heads.
S o l u t i o n :
The total number of trials is 100.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of getting two heads.
The number of times A happens is 36.
Therefore, we have
ii
Let B be the event of getting three heads
The number of times B happens is 12.
Therefore, we have
iii
Let C be the event of getting at least one head.
The number of times C happens is .
Therefore, we have
iv
Let D be the event of getting more heads than tails.
The number of times D happens is .
Therefore, we have
v
Let E be the event of getting more tails than heads.
The number of times E happens is .
Therefore, we have
Q u e s t i o n : 4
1500 families with 2 children were selected randomly and the following data were recorded:
 
Number of girls in a family 0 1 2
Number of families 211 814 475
If a family is chosen at random, compute the probability that it has:
i
No girl
ii
1 girl
iii
2 girls
iv
at most one girl
v
more girls than boys
S o l u t i o n :
The total number of trials is 1500.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of having no girl.
The number of times A happens is 211.
Therefore, we have
ii
Let B be the event of having one girl.
The number of times B happens is 814.
Therefore, we have
iii
Let C be the event of having two girls.
The number of times C happens is 475.
Therefore, we have
iv
Let D be the event of having at most one girl.
The number of times D happens is .
Therefore, we have
v
Let E be the event of having more girls than boys.
The number of times E happens is 475.
Therefore, we have
Q u e s t i o n : 5
In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.
i
he hits boundary
ii
he does not hit a boundary.
S o l u t i o n :
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Page 5


Q u e s t i o n : 1
A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545
Compute the probability for each event.
S o l u t i o n :
The coin is tossed 1000 times. So, the total number of trials is 1000.
Let A be the event of getting a head and B be the event of getting a tail.
The number of times A happens is 455 and the number of times B happens is 545.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 2
Two coins are tossed simultaneously 500 times with the following frequencies of different outcomes:
Two heads: 95 times
One tail: 290 times
No head: 115 times
Find the probability of occurrence of each of these events.
S o l u t i o n :
The total number of trials is 500.
Let A be the event of getting two heads, B be the event of getting one tail and C be the event of getting no head.
The number of times A happens is 95, the number of times B happens is 290 and the number of times C happens is
115.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
Therefore, we have
Q u e s t i o n : 3
Three coins are tossed simultaneously 100 times with the following frequencies of different outcomes:
Outcome: No head One head Two heads Three heads
Frequency: 14 38 36 12
If the three coins are simultaneously tossed again, compute the probability of:
i
2 heads coming up.
ii
3 heads coming up.
iii
at least one head coming up.
iv
getting more heads than tails.
v
getting more tails than heads.
S o l u t i o n :
The total number of trials is 100.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of getting two heads.
The number of times A happens is 36.
Therefore, we have
ii
Let B be the event of getting three heads
The number of times B happens is 12.
Therefore, we have
iii
Let C be the event of getting at least one head.
The number of times C happens is .
Therefore, we have
iv
Let D be the event of getting more heads than tails.
The number of times D happens is .
Therefore, we have
v
Let E be the event of getting more tails than heads.
The number of times E happens is .
Therefore, we have
Q u e s t i o n : 4
1500 families with 2 children were selected randomly and the following data were recorded:
 
Number of girls in a family 0 1 2
Number of families 211 814 475
If a family is chosen at random, compute the probability that it has:
i
No girl
ii
1 girl
iii
2 girls
iv
at most one girl
v
more girls than boys
S o l u t i o n :
The total number of trials is 1500.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of having no girl.
The number of times A happens is 211.
Therefore, we have
ii
Let B be the event of having one girl.
The number of times B happens is 814.
Therefore, we have
iii
Let C be the event of having two girls.
The number of times C happens is 475.
Therefore, we have
iv
Let D be the event of having at most one girl.
The number of times D happens is .
Therefore, we have
v
Let E be the event of having more girls than boys.
The number of times E happens is 475.
Therefore, we have
Q u e s t i o n : 5
In a cricket match, a batsman hits a boundary 6 times out of 30 balls he plays.
i
he hits boundary
ii
he does not hit a boundary.
S o l u t i o n :
The total number of trials is 30.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of hitting boundary.
The number of times A happens is 6.
Therefore, we have
ii
Let B be the event of does not hitting boundary.
The number of times B happens is .
Therefore, we have
Q u e s t i o n : 6
The percentage of marks obtained by a student in monthly unit tests are given below:
 
Unit test: I II III IV V
Percentage of marks obtained: 69 71 73 68 76
Find the probability that the student gets:
i
more than 70% marks
ii
less than 70% marks
iii
a distinction
S o l u t i o n :
The total number of trials is 5.
Remember the empirical or experimental or observed frequency approach to probability.
If n be the total number of trials of an experiment and A is an event associated to it such that A happens in m-trials.
Then the empirical probability of happening of event A is denoted by and is given by
i
Let A be the event of getting more than 70% marks.
The number of times A happens is 3.
Therefore, we have
ii
Let B be the event of getting less than 70% marks.