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.
