You omitted the probabilities for the first contestant.
A contestant on the game show called "A Diamond for your Wife" gets to select 5 diamonds from a box.
The box contains 20 diamonds of which 8 are fake and 12 are real.
Regardless of how many real diamonds the contestant has after his selection, he can only take one home for his wife.
A second contestant then gets to select one diamond from the remaining 15 diamonds from the box.
What is the probability that this second contestant selects a real diamond?
There are: .
Contestant-1 selects 5 diamonds.
There are: . possible outcomes.
There are six possible scenarios:
 Contestant-1 chooses 0 Fake and 5 Reals.
 Contestant-1 choose 1 Fake and 4 Reals.
. . . . . . and so on.