Learn / Library / Probability

Waiting for a marked card

ProbabilityMedium

A deck has $20$ cards, and exactly $3$ of them are marked. The deck is shuffled uniformly and you flip cards one at a time until the first marked card appears. What is the expected number of cards flipped, counting the marked card itself? Give a fraction in lowest terms.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

The marked cards split the unmarked cards into groups; use symmetry for the first group.

Solution

There are $17$ unmarked cards, and the $3$ marked cards cut them into $4$ gaps.
By symmetry, each gap has the same expected size, so the first gap has expected size $\frac{17}{4}$ .
The first gap is exactly the set of unmarked cards flipped before the first marked card.
Add the marked card itself: $\frac{17}{4} + 1 = \frac{21}{4}$ .

Answer: $\frac{21}{4}$

Next question Back to the library