Question

in 6 - card poker, played with a standard 52 - card deck, 52c6, or 20,358,520, different hands are possible. the probability of being dealt various hands is the number of different ways they can occur divided by 20,358,520. 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. the probability is (round to six decimal places as needed.)

Explanation:

Step1: Recall probability formula

The probability of an event $A$ is $P(A)=\frac{n(A)}{n(S)}$, where $n(A)$ is the number of favorable outcomes and $n(S)$ is the total number of outcomes. Here, $n(S) = 20358520$ (total number of 6 - card hands from a 52 - card deck). Let the number of a particular type of hand be $n(A)$. The probability of getting this type of hand is $P=\frac{n(A)}{20358520}$, and the probability of not getting this type of hand is $P(\text{not }A)=1 - P = 1-\frac{n(A)}{20358520}$. Assume the number of a particular type of hand $n(A)$ is known. But if we want a general formula for the probability of not getting a particular hand, we use the complement rule. Since we are not given the number of a particular hand, we assume the probability of getting a particular hand is $p$ and the probability of not getting it is $1 - p$.

Step2: Calculate the probability of non - occurrence

We know that the sum of the probability of an event and its complement is 1. Let's assume the probability of getting the specific hand is $p$. Then the probability of not getting it is $1 - p$. Since we are not given the number of the specific hand, if we assume the number of ways to get the specific hand is $x$, then $p=\frac{x}{20358520}$, and the probability of not getting it is $1-\frac{x}{20358520}=\frac{20358520 - x}{20358520}$.

Answer:

We need to know the number of a particular type of hand to calculate the exact probability. If the number of a particular type of hand is $x$, the probability of not being dealt this type of hand is $\frac{20358520 - x}{20358520}$. If we assume no information about the specific hand, we can only say that if the probability of getting a specific hand is $p$, the probability of not getting it is $1 - p$.