In fivecard poker a straight consists of five cards with adj

In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 10? What is the probability that it will be a straight? What is the probability that it will be a straight flush (all cards in the same suit)?

Solution

31.

a) STRAIGHT WITH HIGH CARD 10

There are 52C5 ways to draw any 5 cards.

As there are 4 suits, there are 4*4*4*4*4 = 1024 ways to choose cards from 6 to 10 to form a straight. However, we subtract 4 straight flushes here, so we just have 1020 ordinary straights.

Thus,

P(straight with high card 10) = 1020/(52C5) = 0.000392465 [answer]

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b) STRAIGHT:

There are 10 ways to choose a high card (5 to A). Thus, from the result above, there are 10*1020 = 10200 straights.

Thus,

P(straight) = 10200/(52C5) = 0.003924647 [answer]

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c)

There are 10 ways to choose a high card, and 4 suits for a straight flush. Hence, there are 10*4 = 40 straight flushes.

Thus,

P(straight flush) = 40/(52C5) = 0.0000153908 [answer]