5 cards are dealt, what is the probability that the fifth one is a king, given that the first 4 are hearts?

It's either 1/12 (4/48) or 1/16 (3/48), depending on whether the four hearts included the king of hearts.

So what is the probability of that event? There are

12 * 11 * 10 * 9 = 11,880

ways to deal four hearts without the king.

There are four possible positions in which the king can appear among four cards, and

12 * 11 * 10 = 1320

ways to deal three other hearts, so that is

4 * 1320 = 5280 ways to deal four hearts including the king.

A quick look at the factors in each of the above leads us to notice that the odds of including the king in the first four hearts were 4 to 9, so the overall probability that the fifth card is a king is

4/48 * 9/13 + 3/48 * 4/13

= (36 + 12) / (48 * 13) = 1/13

So if your gut feeling was to ignore the suits of the first four cards and treat them as independent of the likelihood of a king, you were right!

4cards including a king have been dealt 48 cards including 3 kings remain Pr = 3/48 = 1/16 <-------