Probability Playing Cards

Question

Three cards are drawn from an ordinary deck of playing cards. You flip the last two of them and find that both are spades. What is the probability that also the first card is a spade?

Answer 1

Let event A is "1st card is a spade", event B is "2nd card is a spade", event C is "3rd card is a spade"

You need to find P(A | B and C) = P(A and B and C) / P(B and C).

1. P(A and B and C) = P(A) × P(B | A) × P(C | A and B) = 13/52 × 12/51 × 11/50 = 11/850
2. P(B and C) = P(A and B and C) + P(not A and B and C)

P(not A and B and C) = P(not A) × P(B | not A) × P(C | not A and B) = 39/52 ×13/51 × 12/50 = 39/850

P(B and C) = 11/850 + 39/850 = 50/850

Finally, P(A | B and C) = (11/850) / (50/850) = 11/50 = 0.22.