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# probability

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Two cards are chosen at random from a standard 52-card deck. What is the probability that they are either both hearts or both diamonds or both Aces?

May 21, 2021

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Let H, D, and A be the events that the two cards selected are both hearts, diamonds, or aces, respectively. First note that these events are disjoint; the two cards cannot simultaneously be two aces and two diamonds or two hearts (there is only one ace of diamonds and only one ace of hearts). So

$$p(H\ or\ D\ or\ A)= p(H)+p(D) + p(A)$$

Now

$$p(H)=p(D)=\frac{C(13, 2)}{C(52, 2)}=\frac{1}{17}$$, and

$$p(A)=\frac{C(4, 2)}{C(52, 2)}=\frac{1}{221}$$. So,

$$p(H\ or\ D\ or\ A)=\frac{1}{17}+\frac{1}{17}+\frac{1}{221}=\frac{27}{221}.$$

May 21, 2021