Question

Question: What is the probability that a five-card poker hand contains cards of five different kinds.

Short answer

Answer

The probability that a five-card poker hand contains cards of five different kinds is $P\left(E\right)\approx 51\%$.

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

Here, we have to find the probability that a five-card poker hand contain cards of five different 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:

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

As we know that there a total of fifty-two cards and five cards can be chosen in

${}^{52}{C}_5$.

We are supposed to draw a five-card poker hand contain cards of five different kinds and this can be accomplished as follows:

The first card can be drawn in 52 ways.

The second card can be drawn in 48 ways.

The third card can be drawn in 44 ways.

The fourth card can be drawn in 40 ways.

The fifth card can be drawn in 36 ways.

$$\begin{gathered} \therefore n\left(E\right)=\frac{52\times 48\times 44\times 40\times 36}{5!} \\ P\left(E\right)=\frac{\frac{52\times 48\times 44\times 40\times 36}{5!}}{52C_5} \\ P\left(E\right)=\frac{52\times 48\times 44\times 40\times 36\times 5!\times 47!}{52!\times 5!} \\ P\left(E\right)=\frac{44\times 40\times 36}{51\times 50\times 49} \end{gathered}$$

SolveFurther we get,

$$\begin{gathered} P\left(E\right)=0.50708 \\ P\left(E\right)\approx 51\% \end{gathered}$$

The probability that a five-card poker hand contains cards of five different kinds is $P\left(E\right)\approx 51\%$.