Question

this exercise refers to a poker hand consisting of 5 cards dealt at random from a standard deck of 52 cards. find the probability of getting the specified hand. (round your answer to seven decimal places.) a royal flush (ten, jack, queen, king, and ace of the same suit) the probability of a royal flush is

Explanation:

Step1: Calculate the number of ways to get a royal flush

There are 4 suits (hearts, diamonds, clubs, spades). For each suit, there is exactly 1 way to get a royal flush (ten, jack, queen, king, ace of that suit). So the number of royal - flush hands \(n = 4\).

Step2: Calculate the number of ways to choose 5 cards from a 52 - card deck

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 52\) (total number of cards) and \(r=5\) (number of cards in a hand). So \(C(52,5)=\frac{52!}{5!(52 - 5)!}=\frac{52\times51\times50\times49\times48}{5\times4\times3\times2\times1}=2598960\).

Step3: Calculate the probability

The probability \(P\) of an event is \(P=\frac{\text{Number of favorable outcomes}}{\text{Number of total outcomes}}\). So \(P=\frac{4}{2598960}\approx0.00000154\).

Answer:

\(0.0000015\)