Chapter CSB, Problem 4.22E

To determine

Find the probability that both cards are clubs.

Answer to Problem 4.22E

  $theoretical$

  $\frac{1}{17}.$

Explanation of Solution

Given information:

A hand of $2$ cards is dealt from a standard deck of cards.

Calculation:

The total number of cards in a deck are $52.$

As $2$ cards are drawn from the deck, so, the probability that both cards are clubs is,

  $\begin{array}{l}P\left(bothcardsareclubs\right)=\frac{{}_{13}{C}_2}{{}_{52}{C}_2}\\ =\frac{\frac{13!}{2!\left(11!\right)}}{\frac{52!}{2!\left(50!\right)}}\\ =\frac{78}{1326}\\ =\frac{1}{17}\end{array}$

The probabilities that are based on known characteristics and facts are called theoretical probabilities.

The probabilities that are based on outcomes obtained by conducting an experiment are called experimental probabilities.

An arrangement or selection of objects in which order is not important is called a combination.

Here, found by using the known characteristics and facts as there are $13$ clubs in a deck of $52$ cards.

Hence, the probability is $theoretical$ and the probability $P\left(bothcardsareclubs\right)is\frac{1}{17}.$