Problem

In 7-card poker, played with a standard 52-card deck, ${ }_{52} \mathrm{C}_{7}$, or $133,784,560$, different hands are possible. The probability of being dealt various hands is the number of different ways they can occur divided by $133,784,560$. Shown to the right is the
\begin{tabular}{|c|c|}
\hline \begin{tabular}{c}
Number of Ways the Hand \\
Can Occur
\end{tabular} & Probability \\
\hline 786 & $\frac{786}{133,784,560}$ \\
\hline
\end{tabular}
number of ways a particular type of hand can occur and its associated probability. Find the probability of not being dealt this type of hand.

The probability is $\square$. (Round to six decimal places as needed.)

Answer

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Answer

Rounding this to six decimal places, we find that the probability of not being dealt this type of hand is approximately $\boxed{0.999994}$.

Steps

Step 1 :In a 7-card poker game played with a standard 52-card deck, there are ${ }_{52} \mathrm{C}_{7}$, or $133,784,560$, different possible hands.

Step 2 :The probability of being dealt a particular type of hand is calculated by dividing the number of different ways that hand can occur by the total number of possible hands. In this case, there are 786 ways to get a particular type of hand, so the probability is $\frac{786}{133,784,560}$.

Step 3 :To find the probability of not being dealt this type of hand, we subtract the probability of getting the hand from 1. This gives us $1 - \frac{786}{133,784,560}$.

Step 4 :Calculating this gives us a probability of approximately 0.9999941248825724.

Step 5 :Rounding this to six decimal places, we find that the probability of not being dealt this type of hand is approximately $\boxed{0.999994}$.

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