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Probability Distribution

(a) A Probability Distribution spells out how a total probability of 1 is distributed over several values of a random variable.

(b) Mean of any probability distribution of a random variable is given by Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(c) Variance of a random variable is given by Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced ⇒ Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced ( Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced SD = +Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced)

(d) The probability distribution for a binomial variate 'X' is given by ; P (X = r) = nCr pr qn-r. The recurrence formula Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced, is very helpful for quickly computing P(1), P(2), P(3) etc. if P(0) is known.

(e) Mean of BPD = np ; variance of BPD = npq.

(f) If p represents a persons chance of success in any venture and 'M' the sum of money which he will receive in case of success, then his expectations or probable value = pM

expectations = pM

Geometrical Probability 

The following statements are axiomatic :

(a) If a point is taken at random on a given straight line AB, the chance that it falls on a particular segment PQ of the line is PQ/AB

(b) If a point is taken at random on the area S which includes an area s, the chance that the points falls on σ is σ/S.

Other Definitions Of Probability 

(a) Axiomatic probability : Axiomatic approach is another way of describing probability of an event, in this approach some axioms or rules are depicted to assign probabilities.

Let S be the sample space of a random experiment. The probability P is a real valued function whose domain is the power set of S and range is the interval [0, 1] satisfying the following axioms:

(i) For any event E, P (E) Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced 0

(ii) P(S) = 1

(iii) If E and F are mutually exclusive events, the P (E U F) = P(E) + P(F)

It follows from (iii) that P(φ) = 0

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(b) Empirical probability : A method which can be adopted in the example given above is to throw the dart several times (each throw is a trial) and count the number of times you hit the bull's-eye (a success) and the number of times you miss (a failure). Then an empirical value of the probability that you hit the bull's - eye with any one throw is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

If the number of throws is small this does not give a particular good estimate but for a large number of throws the result is more reliable.

When the probability of the occurrence of an event A cannot be worked out exactly, an empirical value can be found by adopting the approach described above, that is :

(i) making a large number of trials (i.e. set up an experiment in which the event may, or may not, occur and note the outcome)

(ii) counting the number of times the event does occur, i.e. the number of successes,

(iii) calculating the value of Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

The probability of then event A occurring is defined as Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced mean that the number of trials is large (but what should be taken as `large' depends on the problem).

Important Points

(a) If Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced then Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced and Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(b) If Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced where A1, A2, ....... An are mutually exclusive events then P(A) = P(A1) + P(A2) + ...... P(An) = 1

(c) Let A & B are two events corresponding to sample space S then P(S|A) = P(A|A) = 1

(d) Let A and B are two events corresponding to sample space S and F is any other event s.t. P(F) ≠ 0 then Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(e)Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(f)Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

 

Ex 1A, B, C in order cut a pack of cards, replacing them after each cut, on the condition that the first who cuts a spade shall win a prize; find their respective chances.

Sol. Let p be the chance of cutting a spade and q be the chance of not cutting a spade from a pack of 52 cards. Then Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced and Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Now A will win a prize if he cuts spade at 1st, 4th, 7th, 10th turns, etc. Note that A will get a second chance if A, B, C all fail to cut a spade once and then A cuts a spade at the 4th turn.

Similarly he will cut a spade at the 7th turn when A, B, C fail to cut spade twice, etc.

Hence A's chance of winning the prize = Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Similarly B's chance Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

and C's chance Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced of B's chance Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

 

Ex.2 (a) If p and q are chosen randomly from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, with replacement, determine the probability that the roots of the equation x2 + px + q = 0 are real.

(b) Each coefficient in the equation ax2 + bx + c = 0 is determined by throwing at ordinary die. Find the probability that the equation will have equal roots.

Sol. (a) If roots of x2 + px + q = 0 are real, then p2 - 4q Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced 0 ....(i)

Both p, q belongs to set Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced when p = 1, no value of q from S will satisfy (i)

p = 2  ,q = 1 will satisfy, 1 value

p = 3 ,q = 1, 2 ,2 value

p = 4  , q = 1, 2, 3, 4 ,4 value

p = 5 , q = 1, 2, 3, 4, 5, 6, 6 value

p = 6 ,q = 1, 2, 3, 4, 5, 6, 7, 8, 9, ,9 value

For p = 7, 8, 9, 10 all the ten values of q will satisfy.

Sum of these selections in 1 + 2 + 4 + 6 + 9 + 10 + 10 + 10 = 62

But the total number of selections of p and q without any order is 10 × 10 = 100

Hence the required probability is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

(b) Roots equal Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced                 Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced ....(i)

Each coefficient is an integer, so we consider the following cases :

b = 1   , Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced No integral values of a and c

b = 2 , 1 = ac Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced (1, 1)

b = 3 , 9/2 = ac No integral values of a and c

b = 4 ,  4 = ac Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced (1, 4), (2, 2), (4, 1)

b = 5 , 25/2 = ac No integral values of a and c

b = 6 ,9 = ac Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced (3, 3)

Thus we have 5 favourable way for b = 2, 4, 6,

Total number of equations is 6.6.6 = 216 Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced Required probability is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

 

Ex.3 In a test an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is and the probability that he copies the answer is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced. The probability that his answer is correct given that he copied it, is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced. Find the probability that he knew the answer to the question given that he correctly answered it.

Sol. Let A1 be the event that the examinee guesses that answer ; A2 the event that he copies the answer and A3 the event that he knows the answer. Also let A be the event that he answers correctly. Then as given, we have Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

[We have assumed here that the events A1, A2 and A3 are mutually exclusive and totally exhaustive.]

Now Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced (as given)

Again it is reasonable to take the probability of answering correctly given that he knows the answer as 1, that is Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced. We have to find Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced.

By Baye's theorem, we have Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

 

Ex.4 A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events A, B, C are defined as

A = {The first bulb is defective}

B = {The second bulb is non-defective}

C = {The two bulbs are both defective or both non-defective}

Determine whether 

(i) A, B, C are pairwise independent, 

(ii) A, B, C are independent.

Sol. We have Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced; Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced; Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced is the event that first bulb is defective and second is non-defective.    

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & AdvancedProbability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advancedis the event that first bulb is defective and second is non-defective.     

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & AdvancedProbability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Similarly Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced. Thus we have Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced A, B and C are pairwise independent. There is no element in Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & AdvancedProbability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced

Hence A, B and C are not mutually independent.

 

The document Probability Distribution of a Random Variable | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Probability Distribution of a Random Variable - Mathematics (Maths) for JEE Main & Advanced

1. What is a probability distribution of a random variable?
Ans. A probability distribution of a random variable is a function that assigns probabilities to each possible outcome of the random variable. It describes the likelihood of each possible value occurring and provides a summary of the distribution's properties.
2. How is a probability distribution different from a probability density function (PDF)?
Ans. A probability distribution refers to the set of all possible outcomes and their associated probabilities, whereas a probability density function (PDF) is a mathematical function that describes the likelihood of a continuous random variable taking on a particular value within a given range.
3. What are the types of probability distributions commonly used in statistics?
Ans. Some commonly used probability distributions in statistics include the normal distribution, binomial distribution, Poisson distribution, exponential distribution, and uniform distribution. Each distribution has its own characteristics and is used to model different types of data.
4. How can probability distributions be used in real-life applications?
Ans. Probability distributions have numerous real-life applications. For example, they can be used to model stock prices in finance, defect rates in manufacturing, waiting times in queueing systems, and customer arrivals in call centers. By understanding the probability distribution of a random variable, we can make predictions and make informed decisions in various fields.
5. What is the expected value of a random variable and how is it calculated?
Ans. The expected value of a random variable is a measure of the central tendency of its probability distribution. It represents the average value or the long-term average outcome. The expected value is calculated by multiplying each possible outcome by its corresponding probability and summing them up. It provides valuable insights into the behavior of the random variable and can be used for decision-making and risk assessment.
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