Chapter 2: Q. 2.15 (page 49)

If it is assumed that all $(\begin{matrix} 52 \\ 5 \end{matrix})$ poker hands are equally likely, what is the probability of being dealt
$(a)$ a flush? (A hand is said to be a flush if all $5$ cards are of the same suit.)
$(b)$ one pair? (This occurs when the cards have denominations $a, a, b, c, d,$ where $a, b, c,$ and $d$ are all distinct.)
$(c)$ two pairs? (This occurs when the cards have denominations $a, a, b, b, c,$ where $a, b,$ and $c$ are all distinct.)
$(d)$ three of a kind? (This occurs when the cards have denominations $a, a, a, b, c,$ where $a, b,$ and $c$ are all distinct.)
$(e)$ four of a kind? (This occurs when the cards have denominations $a, a, a, a, b$ )

Expert verified

$a) 0.0019 b) 0.4225 c) 0.04754 d) 0.0211 e) 0.0002$

Step by step solution

If it is assumed that all $(\begin{matrix} 52 \\ 5 \end{matrix})$ poker hands are equally likely,

$P (f l u s h) = [(4 C 1) * (13 C 5)] / 52 C 5 = 0.0019$ .

$P (o n e p a i r) = [(13 C 1) * (4 C 2) *) (12 C 3) * (4 C 1) * (4 C 1) * (4 C 1)] / (52 C 5) = 0.4225$ .

$P (t w o p a i r s) = [(13 C 1) * (4 C 2) * (4 C 2) * (11 C 1) * (4 C 1)] / (52 C 5) = 0.04754$ .

$P (t h r e e o f a k i n d) = [(13 Cl) * (4 C 3) * (12 C 2) * (4 Cl) * (4 Cl)] / (52 C 5) = 0.0211$ .

$P (f o u r o f a k i n d) = [(13 Cl) * (4 C 4) * (12 C 1) * (4 Cl)] / (52 C 5) = 0.0002$ .

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