Question

in a five - card poker hand, what is the probability of being dealt exactly one three and no picture cards? click the icon to view a description of a standard deck of playing cards. the probability is \\(\square\\). (round to four decimal places as needed.)

Explanation:

Step1: Determine the total number of five - card hands

The number of combinations of choosing \(r\) items from \(n\) items is given by the formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n!=n\times(n - 1)\times\cdots\times1\). For a standard deck of \(n = 52\) cards and a five - card hand (\(r = 5\)), the total number of five - card hands is \(C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52!}{5!47!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960\).

Step2: Determine the number of ways to get exactly one three and no picture cards

Sub - step 2.1: Number of ways to choose one three

There are 4 threes in a deck. The number of ways to choose 1 three from 4 is \(C(4,1)=\frac{4!}{1!(4 - 1)!}=\frac{4!}{1!3!}=4\).

Sub - step 2.2: Determine the number of non - picture, non - three cards

Picture cards are Jack, Queen, King. There are 3 picture cards in each of the 4 suits, so there are \(3\times4 = 12\) picture cards. The total number of non - picture cards is \(52-12 = 40\). But we have already excluded the 4 threes, so the number of non - picture, non - three cards is \(40 - 4=36\).

Sub - step 2.3: Number of ways to choose 4 non - picture, non - three cards

We need to choose 4 cards from these 36 non - picture, non - three cards. The number of ways is \(C(36,4)=\frac{36!}{4!(36 - 4)!}=\frac{36!}{4!32!}=\frac{36\times35\times34\times33}{4\times3\times2\times1}=58905\).

Sub - step 2.4: Number of favorable hands

By the multiplication principle, the number of hands with exactly one three and no picture cards is the product of the number of ways to choose one three and the number of ways to choose 4 non - picture, non - three cards. So the number of favorable hands \(N = C(4,1)\times C(36,4)=4\times58905 = 235620\).

Step3: Calculate the probability

The probability \(P\) is the number of favorable hands divided by the total number of five - card hands. So \(P=\frac{C(4,1)\times C(36,4)}{C(52,5)}=\frac{4\times58905}{2598960}=\frac{235620}{2598960}\approx0.0907\).

Answer:

\(0.0907\)