<p>Alice knows that <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" /> red cards and <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" /> 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 <img alt="$\frac{m}{n},$" height="33" src="https://latex.artofproblemsolving.com/8/8/8/888fed89e19182447f85b3cb6676548a49395284.png" width="24" /> where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n.$" height="12" src="https://latex.artofproblemsolving.com/5/d/b/5db26f75f6218a382840f0ed34f5d36bcff61d33.png" width="52" /></p>

Question No: 340

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 $m+n.$

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Tags: Probability

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