Chapter 8, Problem 1RE

To determine

The probability of being dealt 5 clubs.

Answer to Problem 1RE

The probability of being dealt 5 clubs is approximately 0.0005.

Explanation of Solution

Given:

An experiment consists a standard 52 card deck.

Formula used:

The formula for number of combinations of n distinct objects taken r at a time is, ${}_n{C}_r=\frac{n!}{\left(n-r\right)!r!}$.

Probability:

If we assume that each simple event in sample space S is as likely to occur as any other, then the probability of an arbitrary event E in S is given by,

$\begin{array}{c}P\left(E\right)=\frac{\text{number of elements in }E}{\text{number of elements in }S}\\ =\frac{n\left(E\right)}{n\left(S\right)}\end{array}$

Calculation:

A standard deck of 52 cards has four 13-card suits: diamonds, hearts, clubs, and spades.

The diamonds and hearts are red, and the clubs and spades are black. Each 13-card suit contains cards numbered from 2 to 10, a jack, a queen, a king, and an ace.

The jack, queen, and king are called face cards.

Let A denote the set of a single deal of 5 club cards, and S denote the set of five cards.

From the 52 cards selecting five cards can be done in ${}_{52}{C}_5$ ways. That is, $n\left(S\right){=}_{52}{C}_5$, and from the 13 clubs cards selecting five clubs cards can be done in ${}_{13}{C}_5$ ways. That is, $n\left(A\right){=}_{13}{C}_5$.

The probability of being dealt 5 clubs is,

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

Further simplified as,

$\begin{array}{c}P\left(A\right)=\frac{47!\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8!}{52\cdot 51\cdot 50\cdot 49\cdot 48\cdot 47!\cdot 8!}\\ =\frac{13\cdot 12\cdot 11\cdot 10\cdot 9}{52\cdot 51\cdot 50\cdot 49\cdot 48}\\ =\frac{154440}{311875200}\\ \approx 0.0005\end{array}$

Thus, the probability of being dealt 5 clubs is approximately 0.0005.