3 1 - 3 2 A jar contains five red balls numbered 1 to 5 and seven green balls numbered 1 to 7 .

Two balls are drawn at random from the jar. Find the following conditional probabilities.

(a) The second ball drawn is red, given that the first is red.

(b) The second ball drawn is red, given that the first is green.

(c) The second ball drawn is even-numbered, given that the first is odd numbered.

(d) The second ball drawn is even numbered, given that the first is even numbered.

(a)

To find:

The conditional probability that the second ball is red, given that the first is red.

Given:

The jar contains five red balls numbered 1 to 5 and seven green balls numbered 1 to 7.

Two balls are drawn from the jar at random.

Approach:

Let E and F be two events in a sample space S. The conditional probability of E given that F is given as,

P(E|F)=P(E∩F)P(F) ……(1)

Let E and F be two independent events in a sample space S then the probability of E and F is P(E∩F)=P(E)P(F) ……(2)

Calculation:

Let E be the event that the first ball drawn is red, F be the event that the second ball drawn is red. Total number of balls is 12, As the both events are independent, the probability P(E) of selecting first red ball from 12 balls is 512 and the probability P(F) of selecting second red ball is also 411

Substitute 512 for P(E), 411 for P(F) in equation (2)

(b)

To find:

The conditional probability that the second ball is red, given that the first is green.

(c)

To find:

The conditional probability that the second ball drawn is even-numbered, given that the first is odd-numbered.

(d)

To find:

The conditional probability that the second ball drawn is even-numbered, given that the first is even numbered.