raetaylor0507
Answered

In 5-card poker, find the probability of being dealt the following hand. Refer to the table. Note that a standard deck of playing cards has 52 cards - 4 suits (clubs, diamonds, hearts, spades), where each suit has 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King).

Four kings

The probability of being dealt four kings is [tex]$\square$[/tex].

(Type an integer or decimal rounded to eight decimal places as needed.)

\begin{tabular}{|l|r|}
\hline \multicolumn{1}{|c|}{ Event E } & \begin{tabular}{c}
Number of \\
Outcomes \\
Favorable to E
\end{tabular} \\
\hline Royal flush & 4 \\
\hline Straight flush & 624 \\
\hline Four of a kind & 3744 \\
\hline Full house & 5108 \\
\hline Flush & 10,200 \\
\hline Straight & 54,912 \\
\hline Three of a kind & 123,552 \\
\hline Two pairs & [tex]$1,098,240$[/tex] \\
\hline One pair & [tex]$1,302,540$[/tex] \\
\hline No pair & [tex]$2,598,960$[/tex] \\
\hline Total & \\
\hline
\end{tabular}



Answer :

GinnyAnswer
To determine the probability of being dealt four kings in a 5-card poker hand, we'll follow these steps:

### Step-by-Step Solution:

1. Understand and Set Up the Problem:
- We are working with a standard 52-card deck.
- We need to find the probability of drawing a hand with exactly four kings.

2. Calculate Favorable Outcomes:
- First, determine the number of ways to choose 4 kings out of the 4 available in the deck. There is only one way to do this (since all 4 kings must be chosen).
- Next, we choose the remaining 1 card from the rest of the deck. There are [tex]\(52 - 4 = 48\)[/tex] cards left to choose from, and we need to choose 1 card. The number of ways to do this is 48.

Therefore, the total number of favorable outcomes where the hand contains exactly four kings is given by:
[tex]\[ \text{Favorable Outcomes} = 1 \times 48 = 48 \][/tex]

3. Calculate Total Possible Outcomes:
- The total number of 5-card hands that can be dealt from a 52-card deck is found using the combination formula [tex]\(\binom{n}{k}\)[/tex], which represents the number of ways to choose [tex]\(k\)[/tex] items from [tex]\(n\)[/tex] items without regard to order.
- We calculate the total number of 5-card hands from the 52-card deck:
[tex]\[ \binom{52}{5} = \frac{52!}{5!(52-5)!} = 2,598,960 \][/tex]

4. Compute the Probability:
- The probability [tex]\(P\)[/tex] of being dealt four kings in a 5-card hand is the number of favorable outcomes divided by the total number of possible outcomes:
[tex]\[ P(\text{Four Kings}) = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{48}{2,598,960} \][/tex]
- Simplifying the fraction gives:
[tex]\[ P(\text{Four Kings}) = 1.846892603195124 \times 10^{-5} \][/tex]
- Therefore, the probability rounded to eight decimal places is:
[tex]\[ 0.00001847 \][/tex]

### Final Answer:
[tex]\[ \boxed{0.00001847} \][/tex]

This is the probability of being dealt a hand containing exactly four kings in a standard 5-card poker hand.