Question

Question: What is the probability that a five-card poker hand contain the two of diamonds and three of spades?

Short answer

Answer

The probability that a five-card poker hand contain the two of diamonds and three of spades is$P\left(E\right)=0.0075.$$P\left(E\right)=0.0075.$

Step-by-step solution

Explanation

As per the problem we have been asked to find the probability that a five card poker hand contain the two diamonds and three of spades.

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.

Two diamonds and three of spades can be selected in the poker in $C503$.

Calculation

Now substitute the values in the given formula we get

$$\begin{gathered} P\left(E\right)=C503C525 \\ \Rightarrow P\left(E\right)=50!3!47!52!5!47! \\ \Rightarrow P\left(E\right)=50!3!47!\times 5!47!52! \\ \Rightarrow P\left(E\right)=5663\Rightarrow P\left(E\right)=0.0075 \end{gathered}$$

The probability that a five-card poker hand contain the two of diamonds and three of spades is$P\left(E\right)=0.0075.$