Expected number of jacks drawn given that you draw cards till you draw all 4 kings?

Question

I don't understand how to solve this. Basically define J as our random variable such that J: {0,1,2,3,4}. To solve this, we need to know the probability of getting e.g., 0 jacks given that we draw cards till we get all 4 kings. I don't understand how to compute this probability. I appreciate hints.

Answer 1

Hint. Imagine taking the eight jacks and kings out of the deck, preserving their order. How likely is it that of these eight cards, the first four cards are kings? There is only one such arrangement, so the probability is

$$ p_0 = \frac{1}{\binom{8}{4}} = \frac{1}{70} $$

Now, how likely is it that four of the first five (including the fifth) are kings? How likely is it that four of the first six (including the sixth) are kings? Etc.

Answer 2

Hint: However the cards fall they are arranged in the order $xxxKxxxKxxxKxxxKxxx$ where $xxx$ stands for any number of cards from zero to forty eight. The remaining cards are equally likely to be in any of the five $xxx$ slots. What is the average number of jacks per slot?