You are walking on the street and notice a guy gambling with some other guys and you start to watch.

The game principle is pretty easy. There is a standard 52-card deck and it is being shuffled every time before the game begins. You take five random cards and if at least two of them has the same rank (2,3..., J,Q,K,A) you win, otherwise you lose.

In the long run, you should play or not? (with probabilities of course)


The probability of the first two cards not matching is $\frac{48}{51}$. Given them not matching, the probability of the next card matching neither is $\frac{44}{50}$, and so on.

Our total probability of no pairs, then, is $\frac{48 \times 44 \times 40 \times 36}{51\times 50\times 49\times 48}=.507$

Your odds of losing are slightly better than half, and you should not take the bet.

  • $\begingroup$ you solved it pretty fast :) thanks. $\endgroup$ – Oray Jan 20 '16 at 13:28
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    $\begingroup$ You also shouldn't take the bet because it's a street gambler and you're going to get fleeced, but the OP said to ignore that element. $\endgroup$ – frodoskywalker Jan 20 '16 at 13:30

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