Suppose we had a normal 52 count deck of cards. If two cards are randomly picked, what is the probability that both are Hearts given at least one Heart was chosen.
I think the answer would be $ \frac{12}{51}$ because there are twelve Hearts left after one is chosen with 51 total cards, but I might be approaching this problem the wrong way. Any idea on how I should think about it, or what I did wrong?
Best Answer
There are 3 possibilities to consider:
The chance of #1 is $\frac{39\cdot 38}{52\cdot 51}$
The chance of #2 is $\frac{2\cdot 39\cdot 13}{52\cdot 51}$
The chance of #3 is $\frac{13\cdot 12}{52\cdot 51}$
You can confirm that the probabilities of #1, #2, and #3 add to 1.
#2 and #3 represent cases that at least one heart was chosen. Therefore chance of two hearts chosen given that at least one was chosen are:
$\frac{13\cdot 12}{2\cdot 39\cdot 13+13\cdot 12} = \frac2{15} \approx 13.3\%$