Chapter 9.7, Problem 45E

a.

To determine

To find: the probability that the card is an even-numbered card.

Answer to Problem 45E

  $\frac{5}{13}$

Explanation of Solution

Given:

One card is drawn at random from a standard deck of 52 playing cards.

Let E be the event that the card is an even-numbered card.

The even numbered cards in the deck are 2, 4, 6, 8 and 10. That is total 5 even numbered card are there in each of the 4 suits.

So,

  $\begin{array}{c}n\left(E\right)=5\times 4\\ =20\end{array}$

Total number of cards is 52. So,

  $n\left(S\right)=52$

Thus, the probability of that the card is an even-numbered card is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{20}{52}\\ =\frac{5}{13}\end{array}$

b.

To determine

To find: the probability that the card is a heart or a diamond.

Answer to Problem 45E

  $\frac{1}{2}$

Explanation of Solution

Given:

One card is drawn at random from a standard deck of 52 playing cards.

Let E be the event that the card is an even-numbered card.

There are 13 cards of heart and 13 cards of diamond

So,

  $\begin{array}{c}n\left(E\right)=13+13\\ =26\end{array}$

Total number of cards is 52. So,

  $n\left(S\right)=52$

Thus, the probability of that the card is a heart or a diamond is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{26}{52}\\ =\frac{1}{2}\end{array}$

c.

To determine

To find: the probability that the card is a nine or a face card.

Answer to Problem 45E

  $\frac{4}{13}$

Explanation of Solution

Given:

One card is drawn at random from a standard deck of 52 playing cards.

Let E be the event that the card is a nine or a face card.

There are 4 cards in the deck which are 9 and of 3 face cards in each of the four suits.

So,

  $\begin{array}{c}n\left(E\right)=4+3\times 4\\ =4+12\\ =16\end{array}$

Total number of cards is 52. So,

  $n\left(S\right)=52$

Thus, the probability of that the card is a nine or a face card is:

  $\begin{array}{c}P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}\\ =\frac{16}{52}\\ =\frac{4}{13}\end{array}$