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# Perhaps the most common error in ranking poker hands involves confusing the order of a straight and a flush. A straight is 5 cards with cons

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Perhaps the most common error in ranking poker hands involves confusing the order of a straight and a flush. A straight is 5 cards with consecutive rank, like 45678. The cards rank A,2,3,4,5,6,7,8,9,10,J,Q,K,A. Note that an ace ("A") can be used as high (as in 10JQKA) or as low (as in A2345), but it can't be used as both high and low in the same hand. (That is, you can't use an ace to loop around. KA234 does not count as a straight.) A flush is 5 cards of the same suit. If 5 cards are drawn from the deck at random (without replacement) to form a hand, which hand is more likely: a straight or a flush? Enter straight or flush as your answer, or enter both if they're equally likely.

Guest Apr 3, 2015

#1
+80983
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Probability of a straight.....There are 10 ways to do this within the ranks. {The ace can be the high or low card.} And within each rank, we are choosing any 1 of 4 cards.  Note that we are including the straight flushes and royal flushes, here.

So.....there are C(10,1)*[C(4,1)]^5  = 10,240 possibilities

For the flush, we want to choose any 5 of the 13 ranks. And we have 4 ways to make each of these flushes.....all diamonds, all spades, all hearts or all clubs.  Here again, we are including the royal and straight flushes.

So......there are  C(13, 5) * C(4,1) = 5148 possibilities

Thus....the straight is almost twice as likely to occur......

CPhill  Apr 3, 2015
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#1
+80983
+5

Probability of a straight.....There are 10 ways to do this within the ranks. {The ace can be the high or low card.} And within each rank, we are choosing any 1 of 4 cards.  Note that we are including the straight flushes and royal flushes, here.

So.....there are C(10,1)*[C(4,1)]^5  = 10,240 possibilities

For the flush, we want to choose any 5 of the 13 ranks. And we have 4 ways to make each of these flushes.....all diamonds, all spades, all hearts or all clubs.  Here again, we are including the royal and straight flushes.

So......there are  C(13, 5) * C(4,1) = 5148 possibilities

Thus....the straight is almost twice as likely to occur......

CPhill  Apr 3, 2015

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