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Math Help - Playing cards problem

  1. #1
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    Playing cards problem

    Draw 3 cards from a standard randomized 52-card deck.

    What is the probability all three cards are the same suit?
    What is the probability the three cards are composed of two different suits?
    What is the probability the three cards are composed of three different suits?

    I got an A in my college-level probability class but for some reason I can't seem to get this one right
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  2. #2
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    Re: Playing cards problem

    For the first question, do we care what the first card we pick is? Then given the first card, how many choices do we have for the second and then the third.

    For the remaining two, thinking about Combinations from probability. There are 52 choose 3 possible combinations. To get two different suits we just need to make sure that we choose 2 from the same suit. Can you see where to go from here?

    For the last question, again do we care what the first card we pick is? Then once we choose the first we have a limited number of choices for the second. Then after choosing the first two, we have further limited choices for the third. So we can pick any of the 52 out of 52 for the first. Then we need a different suit, i.e. one card out of the 39 left which are different suits. And now how many can we choose for the third?
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  3. #3
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    Re: Playing cards problem

    another way to approach 2:

    again, we can pick any card for the first card. we can also pick any card for the second card.

    there is a 12/51 chance that the 2nd card is the same suit as the first card. there is a 39/51 chance the 2nd card is not the same suit.

    if the first 2 cards are of the same suit, there is 39/50 chance the 3rd one doesn't match.

    if the first 2 cards are not the same suit, there is a 24/50 chance the 3rd one will be of one of their suits. this gives:

    (12/51)(39/50) + (39/51)(24/50) = 468/2550 + 936/2550 = 1404/2550 = 702/1275 = 234/425
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