Two Red Cards

Question

You draw two cards from a standard 52-card deck without replacement. What is the probability both are red?

Accepted Answer

Draw sequentially and use the chain rule: P(R_1 cap R_2) = P(R_1),P(R_2 mid R_1) = frac2652 cdot frac2551 = frac25102 approx 0.245. After the first red card leaves the deck, only 25 of the remaining 51 cards are red, without-replacement draws are negatively correlated. Approach 2, counting fracbinom262binom522 = frac3251326 = frac25102. Same number, and a good consistency check. Notice it is slightly less than frac14, the with-replacement answer.

Two Red Cards

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