Question

A box contains 3 cards: one white on both sides, one black on both sides and one white on one side and black on the other. A card is drawn at random without replacement.

a. Find the probability that the second side is white if the first side is white.
b. Find the probability that the second side is black if the first side is black.
c. Find the probability that the second side is the same color as the first side if the color of the first side is known.

Answer 1

a. Sample space: { {w/w},{b/b},{w/b} }, only three possible events

P(2nd side is white if 1st side is white}
= P( both sides are white) / P(one side is white), Baye's Rule P(a and b)/P(b)
= P({w/w})/P({w/w} or {w/b}), P(white and white)/P(white), Baye's Theorem
P({w/w}) = 1/3,P({w/b}) = 1/3, P({w/w} or {w/b}) = 2/3

So P(2nd side white given 1st side white) = (1/3)/(2/3) = 3/(2*3) = 3/6 = 1/2
Ans. 1/2

b. (worked similar to a above or use symmetry of a to get answer with no calculations)

c. P(second side same color as the first side if one side known)
= P(other side black given one side is black OR
other side is white given one side white)
= 1 - P(other side not black given one side black AND
other side not white given other side white), complement is easier to compute.
= 1 - (1/2)*(1/2), using complementary results of parts a and b above
= 1 - 1/4
= 3/4

Answer 2

Thank you very much!! Your explanations are extremely helpful and I really appreciate it!