Chapter 9.7, Problem 23E

To determine

To calculate: The probability that the card is a red face card when a card is chosen at random from a deck of 52 cards.

Answer to Problem 23E

The probability that the card drawn from a deck of 52 playing cards is a red face card is $\frac{3}{26}$ .

Explanation of Solution

Given information:

A card is chosen at random from a deck of 52 cards.

Formula used:

The probability of an event is the ratio of number of possible outcomes of an event E denoted by $n\left(E\right)$ and the sample space of equally likely outcome denoted by $n\left(S\right)$ . Mathematically it is denoted by, $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$ .

Calculation:

Consider that acard is chosen at random from a deck of 52 cards.

In a deck of cards there are 12 face cards.

There are 4 suits and each suit has 3 face cards.

There are 2 red suits. So, 6 red face cards.

Recall that the probability of an event is the ratio of number of possible outcomes of an event E denoted by $n\left(E\right)$ and the sample space of equally likely outcome denoted by $n\left(S\right)$ . Mathematically it is denoted by, $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$ .

Denote the event E as drawn card is a red face card. Sample space is 52.

So, $n\left(E\right)=6$ and $n\left(S\right)=52$ .

Probability that a card drawn from a deck of 52 playing card is a red face card is

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{6}{52}\\ =\frac{3}{26}\end{array}$

Thus, the probability that the card drawn from a deck of 52 playing cards is not a face card is $\frac{3}{26}$ .