Question

Question: What is the probability that a five-card poker hand does not contain the queen of hearts?

Short answer

Answer

The probability that a five-card poker hand does not contains the queen of hearts is$P\text{'}\left(E\right)=1-P\left(E\right)=4752$

Step-by-step solution

Given

As per the problem we have been asked to find the probability that a five card poker hand does not contain the queen of hearts.

Explanation

If S represents the sample space and E represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

$P\left(E\right)=n\left(E\right)n\left(S\right)$

Five cards can be chosen out of $52$cards in $C525$ways.

There is one queen of heart and $51$other cards. So, if queen of hearts is selected then the rest four can be selected in $C514$ways.

Calculation

Now substitute the values in the given formula we get

$P\left(E\right)=C514C525\Rightarrow P\left(E\right)=51!4!47!52!5!47!\Rightarrow P\left(E\right)=51!4!47!\times 5!47!52!\Rightarrow P\left(E\right)=552$

The probability that a five-card poker hand contains the queen of hearts is

.$P\left(E\right)=552$

The probability that a five-card poker hand does not contains the queen of hearts is .

$P\text{'}\left(E\right)=1-P\left(E\right)=4752$