Math Gold Medalist

Lor

2023 AIME I 

Problem 6

Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$


Expected Value

Small Example

 

red = r , blue = b

Calculate expected values of the following cases:

r

rr

rrr

rb

rrb

rrbb

rrrb

rrrbb

rrrbbb

.

   Solution