Question

Question: To determineThe probability that a five-card poker hand contains a straight flush, that is, five cards of the same suit of consecutive kinds.

Short answer

Answer

The probability that a five-card poker hand contains a straight, that is, five cards that have consecutive kinds is$P\left(E\right)=0.00394$ .

Step-by-step solution

 Given  

A pack of cards. In a pack of cards there are $52$cards.

The Concept of Probability

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

$\mathit{P}\mathbf{\left(}\mathit{E}\mathbf{\right)}\mathbf{=}\frac{\mathbf{n}\mathbf{\left(}\mathbf{E}\mathbf{\right)}}{\mathbf{n}\mathbf{\left(}\mathbf{S}\mathbf{\right)}}$

Determine the probability

As per the problem we have been asked to find the probability that a five-card poker hand contain contains a contains a straight, that is, five cards that have consecutive kinds.

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:

$P\left(E\right)=\frac{n\left(S\right)}{n\left(S\right)}$.

As we know that there a total of fifty-two cards and five cards can be chosen in${}^{52}{C}_5$. and so, we have$n\left(S\right){=}^{52}{C}_5$.

We are supposed to draw a five-card poker hand contains a straight, that is, five cards that have consecutive kinds this can be accomplished as follows:

There are four cards of each kind

The first one can be selected in ${}^4{C}_1$ways.

The second one can be selected in ${}^4{C}_1$ways.

The third one can be selected in ${}^4{C}_1$ways.

The fourth one can be selected in ${}^4{C}_1$ways.

The fifth one can be selected in ${}^4{C}_1$ways.

$$\begin{gathered} P\left(E\right)=\frac{10x^4{C}_1{x}^4{C}_1{x}^4{C}_1{x}^4{C}_1{x}^4{C}_1}{{}^{52}{C}_5} \\ P\left(E\right)=\frac{10\times 4^5}{\frac{51!}{471\times !!}} \\ P\left(E\right)=\frac{10240}{2598960} \\ P\left(E\right)=0.00394 \end{gathered}$$

The probability that a five-card poker hand contains a straight, that is, five cards that have consecutive kinds is $P\left(E\right)=0.00394$.