Question

Consider a box with three tickets numbered one to three. Suppose a coin is tossed. If the coin toss results in a head, then two tickets are drawn from the box with replacement. If the coin toss results in a tail, then two tickets are drawn from the box without replacement. Let A denote the event that the coin toss is a head. Let B be the event that the sum of the tickets drawn is .

(1) Describe the sample space for this experiment.

(2) Assign probabilities to each element of the sample space.

(3) Find P(A) and P(B).

(4) Find P(A|B).

Answer 1

1) Sample space, S = {H11, H12, H13, H22, H23, H33, T12, T13, T23}

2) P(H11) = 1/2 x 1/3 x 1/3 = 1/18

P(H12) = 1/2 x 2 x 1/3x1/3 = 1/9

P(H13) = 1/9

P(H22) = 1/18

P(H23) = 1/9

P(H33) = 1/18

P(T12) = 1/2 x 2 x 1/3 x 1/2 = 1/6

P(T13) = 1/6

P(T23) = 1/6

3) P(A) = 1/2

Let B be the event that sum of the tickets drawn is 4 (4 is an assumption here as it is not given in question)

P(B) = P(H13) + P(H22) + P(T13)

= 1/9 + 1/18 + 1/6

= 1/3

4) P(A|B) = P(A&B)/P(B)

= [P(H13) + P(H22)]/P(H)

= (1/9 + 1/18)/0.5

= 1/3