Conditional Probability

Let us consider the random experiment of throwing a die. Let A be the event of getting an odd number on the die.

\ S = {1, 2, 3, 4, 5, 6} and A = {1, 3, 5}

Let B = {2, 3, 4, 5, 6}. If, after the die is thrown, we are given the information, that the event B has occurred, then the probability of event A will no more be 1/2, because in this case, the favourable cases are two and the total number of possible outcomes will be five and not six. The probability of event A, with the condition that event B has happened will 2/5. This conditional probability is denoted as P(A/B). Let us define the concept of conditional probability in a formal manner.

Let A and B be any two events associated with a random experiment. The probability of occurrence of event A when the event B has already occurred is called the conditional probability of A when B is given and is denoted as P(A/B). The conditional probability P(A/B) is meaningful

By definition,

P(A/B) = Probability of occurrence of event A when the event B as already occurred.

Remark 1:

Remark 2:

If A and B are mutually exclusive events, then

\ If A and B are mutually exclusive events, then A/B and B/A are impossible events.

For an illustration, let us consider the random experiment of throwing two coins.

\ S = {HH, HT, TH, TT}

Let A = {HH, HT}, B = {HH, TH} and C = {HH, HT, TH}

A/B is the event of getting A with the condition that B has occurred.

Remark 3:

We know that for the events A and B,

If B = S then

= P (A)

Remark 4:

If A = B

Remark 5:

From the formula of conditional probabilities, we have

Equation (2) and equation (3) are known as multiplication rules of probability for any two events A and B of the same sample space.

Remark 6:

We know that two events are independent if the occurrence of one does not effect the occurrence of other. If A and B are independent events

P (A/B) = P (A) and P (B/A) = P (B)

\ The multiplication rule for the independent events A and B is given by

Remark 7:

So far, we have assumed that the elementary events are equally likely and we have used the corresponding definition of probability. However the same definition of conditional probability can also be used when the elementary events are not equally likely. This will be clear from the following example.

Suppose a die is tossed. Let B be the event of getting a perfect square.

The die is so constructed that the event numbers are twice as likely to occur as the odd numbers.

Let us find the probability of B given A, where A is the event getting a number greater than 3 while tossing the die.

S = {1, 2, 3, 4, 5, 6}

If probability of getting an odd number is x, the probability of getting an even number is 2x.

Since P (S) = 1

x + 2x + x + 2x + x + 2x = 1

9x = 1

A = {4, 5, 6}

Example 1:

A card is drawn from an ordinary deck and we are told that it is red, what is the probability that the card is greater than 2 but less than 9.

Suggested answer:

Let A be the event of getting a card greater than 2 but less than 9.

B be the event of getting a red card. We have to find the probability of A given that B has occurred. That is, we have to find P (A/B).

In a deck of cards, there are 26 red cards and 26 black cards.

\ n(B) = 26

Among the red cards, the number of outcomes which are favourable

Example 2:

A pair of dice is thrown. If it is known that one die shows a 4, what is the probability that

a) the other die shows a 5

b) the total of both the die is greater than 7

Suggested answer:

Let A be the event that one die shows up 4. Then the outcomes which are favourable to A are

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (1, 4), (2, 4), (3, 4),

(5, 4), (6, 4)

(a) Let B be the event of getting a 5 in one of the dies. Then the outcomes which are favourable to both A and B are (4, 5), (5, 4)

(b) Let C be the event of getting a total of both the die greater than 7.

The out-comes which are favourable to both C and A.

(4, 4), (4, 5), (4, 6), (5, 4), (6, 4)

\ n (C) = 5

Note that in the above example P (B) and P (B/A) are different.

Similarly P (C) and P (C/A) are different.