Explanation Given Information: In poker, a player is dealt 5 cards at random, from a standard deck of 52 cards. A straight flush is when five cards of the same suit with consecutive denominations. A straight has five cards with consecutive denominations but not all of the same suit. Formula used: The probability of an event E , for sample space S is P ( E ) = n ( E ) n ( S ) According to the law of combination, the combination, C ( n , r ) , is solved as, C ( n , r ) = n ! r ! ( n − r ) ! According to the rule for factorials, an n factorial is solved as, n ! = n ( n − 1 ) ... ( 3 ) ( 2 ) ( 1 ) Calculation: Consider the event that a straight is dealt to be E . Total number of cards in a deck are 52. Apply the formula for combination to find the total number ways to draw 5 cards from a deck of 52 by substituting 52 for n and 5 for r , n ( S ) = C ( 52 , 5 ) = 52 ! 5 ! ( 52 − 5 ) ! = 52 ! 5 ! 47 ! Simplify using the property of factorials, n ( S ) = 52 ⋅ 51 ⋅ 50 ⋅ 49 ⋅ 48 ⋅ 47 ! ( 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 ) ⋅ 47 ! = 52 ⋅ 17 ⋅ 10 ⋅ 49 ⋅ 6 = 2598960 There are 13 denominations. A straight has five cards with consecutive denominations. To find a straight, the starting denomination is chosen from A to 10, since, after 10, five consecutive numbers cannot be chosen. Thus, there are 10 choices for the first denomination. The next 4 denominations are selected in 4 5 ways. Thus, the total number of straights is, n ( Straight ) = 10 ⋅ 4 5 = 10 ⋅ 1024 = 10240 The number of straights also includes the consecutive denominations with same suit and cards with K, A, 10, J and Q, that is, a straight flush or royal flush

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Author: Costenoble

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Chapter 8.4, Problem 24E

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To calculate: The probability of a straight, that is, five cards with consecutive denominations but not all of the same suit.

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Chapter 8.4, Problem 23E

Chapter 8.4, Problem 25E

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The probability of a straight, that is, five cards with consecutive denominations but not all of the same suit.

Solution Summary: The author calculates the probability of a straight, that is, five cards with consecutive denominations but not all of the same suit.

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To calculate: The probability of a straight, that is, five cards with consecutive denominations but not all of the same suit.

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