Chapter 8, Problem 1RE

In a single deal of $5$ cards from a standard 52-card deck, what is the probability of being dealt $5$ clubs?

To determine

To calculate:The probability of drawing $5$ clubs in a single deal of $5$ cards drawn from a standard deck of $52$ cards.

Answer to Problem 1RE

The probability of drawing $5$ clubs is $0.0005$ approximately.

Explanation of Solution

Calculation:

Let $E$ be the event of drawing $5$ clubs. The total number of outcomes is $n\left(S\right)=C{}_5$.

The number of club cards in a deck is $13$ out of which $5$ are drawn.

Compute the number of ways in which $5$ club cards are selected.

$n\left(E\right)=C{}_5$

Compute the probability of event $E$ by dividing the number of outcomes of an event by number of outcomes of sample space.

$\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{C{}_5}{C{}_5}\\ =\frac{\frac{13!}{5!8!}}{\frac{52!}{5!47!}}\\ =\frac{13!}{5!8!}\times \frac{5!47!}{52!}\end{array}$

Further solve as,

$\begin{array}{c}P\left(E\right)=\frac{13\times 12\times 11\times 10\times 9\times 8!}{5!8!}\times \frac{5!47!}{52\times 51\times 50\times 49\times 48\times 47!}\\ =\frac{11\times 3}{4\times 17\times 5\times 49\times 4}\\ =\frac{33}{66640}\\ \approx 0.0005\end{array}$

Therefore, the probability of drawing $5$ clubs is $0.0005$ approximately.