Question

A deck of 52 cards has 2 cards missing. 3 cards are drawn at random and are found to be hearts. What is the probability that all the missing cards are hearts?

Answer 1

P(missing cards were all hearts | card drawn was a heart) =

P(card drawn was a heart | missing cards were all hearts) * P(missing cards were all hearts) / P(card drawn was a heart)

Now we can break these parts down step by step.

A) P(card drawn was a heart | missing cards were all hearts)

If this missing cards were all hearts, there would be 10 remaining hearts in the deck of 49. So the probability is 10/49.

B) P(missing cards were all hearts)

Since there are 13 hearts in the deck originally, that means there are 13 possibilities for the first card, 12 for the second, and 11 for the third. So this is simply:

(13/52)(12/51)(11/50)

= .0129

C) P(card drawn was a heart)

This is similar to A) but the trick here is that this is asking for the general probability of drawing 1 heart from a full deck. So the odds are simply 13/52 = 1/4.