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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

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