Expected number of cards to pick up

Question

Suppose you have a well-shuffled pile of cards: the King of Clubs, Queen of Clubs, Jack of Clubs, and 10 of Clubs. You start picking cards from the top until both the King and Queen are picked up. What is the expected number of cards you have to pick?

Would this be 2/12(2) + 4/12(2+E(x)) + 1/2(E(x)+1)

which simplifies to x/6 - 3/2

making x = 9

Sorry, are you picking from a whole deck or just those four cards? The way you phrased it is not clear. — scoopfaze, Nov 26 '19 at 00:36
I asked because I'm confused how the expectation could be larger than the number of cards you have. — scoopfaze, Nov 26 '19 at 01:25

score 1 · Accepted Answer · answered Nov 26 '19 at 01:16

The K,Q could be the first two cards picked.
The probability of that is $\frac {1}{{4\choose2}}$

$P(X=2) = \frac 16$

If you pick 3 cards there is a $\frac 12$ probability that the 4th card is a king or a queen. Meaning you will need to pick the 4th card.

$P(X=4) = \frac 12$

$P(X=3) = 1 - P(X=2) - P(X=4) = \frac 13$

$E[X] = \frac 16\times 2 + \frac 13\times 3 + \frac 12\times 4 = \frac {10}3$

Expected number of cards to pick up

1 Answers1