Question

in 7-card poker, played with a standard 52-card deck, \\(_{52}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 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. \\(\

$$\begin{tabular}{|c|c|} \\hline number of ways the hand can occur & probability \\\\ \\hline 5108 & \\(\\dfrac{5108}{133,784,560}\\) \\\\ \\hline \\end{tabular}$$

\\) the probability is \\(\square\\). (round to six decimal places as needed.)

Explanation:

Step1: Recall Complementary Probability

The probability of an event not occurring is \( 1 - \) the probability of the event occurring. Let \( P(E) \) be the probability of being dealt the hand, so \( P(\text{not } E) = 1 - P(E) \).

Step2: Substitute the Given Probability

We know \( P(E)=\frac{5108}{133784560} \). So \( P(\text{not } E)=1 - \frac{5108}{133784560} \). First, calculate \( 1=\frac{133784560}{133784560} \), then subtract: \( \frac{133784560 - 5108}{133784560}=\frac{133779452}{133784560} \).

Step3: Simplify and Round

Divide \( 133779452 \) by \( 133784560 \). Using a calculator, \( \frac{133779452}{133784560}\approx0.999962 \) (rounded to six decimal places).

Answer:

\( 0.999962 \)