Chapter 4, Problem 24CQ

Selecting a Card When a card is drawn from a deck, find the probability of getting

a. A diamond

b. A 5 or a heart

c. A 5 and a heart

d. A king

e. A red card

To determine

To obtain: The probability of getting a diamond.

Answer to Problem 24CQ

The probability of getting a diamond is $\frac{1}{4}$.

Explanation of Solution

Calculation:

In ordinary deck of cards there are 4 suits. They are hearts, clubs, diamonds and spades. In each suite there are 13 cards. In 13 cards, nine cards are numbers from 2 to 10 and remaining four cards are king, queen, ace and jack cards.

The total number of outcomes is 52.

Let event A denote getting a diamond.

The possible number of outcomes for getting diamonds is 13.

That is, there are 13 outcomes for getting an event A.

The formula for probability of event A is,

$P\left(A\right)=\frac{\text{Number of outcomes in }A}{\text{Total number of outcomes in the sample space}}$

Substitute 13 for ‘Number of outcomes in A’ and 52 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(A\right)=\frac{13}{\text{52}}\\ =\frac{1}{4}\end{array}$

Thus, the probability of getting a diamond is $\frac{1}{4}$.

To determine

To obtain: The probability of getting a 5 or a heart.

Answer to Problem 24CQ

The probability of getting a 5 or a heart is $\frac{4}{13}$.

Explanation of Solution

Calculation:

Let event B denote getting a 5.

The possible number of outcomes for getting a 5 is 5.

That is, there are 4 outcomes for getting an event B.

The formula for probability of event B is,

$P\left(B\right)=\frac{\text{Number of outcomes in }B}{\text{Total number of outcomes in the sample space}}$

Substitute 4 for ‘Number of outcomes in B’ and 52 for ‘Total number of outcomes in the sample space’,

$P\left(B\right)=\frac{4}{\text{52}}$

Let event C denote getting a heart.

The possible number of outcomes for getting a heart is 13.

That is, there are 13 outcomes for getting an event C.

The formula for probability of event C is,

$P\left(C\right)=\frac{\text{Number of outcomes in }C}{\text{Total number of outcomes in the sample space}}$

Substitute 13 for ‘Number of outcomes in C’ and 52 for ‘Total number of outcomes in the sample space’,

$P\left(C\right)=\frac{13}{\text{52}}$

The formula for probability of event A and B is,

$P\left(B\text{ and }C\right)=\frac{\text{Number of outcomes in }B\text{ and }C}{\text{Total number of outcomes in the sample space}}$

Substitute 1 for ‘Number of outcomes in B and C’ and 52 for ‘Total number of outcomes in the sample space’,

$P\left(B\text{ and }C\right)=\frac{1}{52}$

Addition Rule:

The formula for probability of getting event B or event C is,

$P\left(B\text{ or }C\right)=P\left(B\right)+P\left(C\right)-P\left(B\text{ and }C\right)$

Substitute $\frac{4}{\text{52}}$ for $P\left(B\right)$, $\frac{13}{\text{52}}$ for $P\left(C\right)$ and $\frac{1}{\text{52}}$ for $P\left(B\text{ and }C\right)$

$\begin{array}{c}P\left(B\text{ or }C\right)=\frac{4}{52}+\frac{13}{52}-\frac{1}{52}\\ =\frac{4+13-1}{52}\\ =\frac{16}{52}\\ =\frac{4}{13}\end{array}$

Thus, the probability of getting a 5 or a heart is $\frac{4}{13}$.

To determine

To obtain: The probability of getting a 5 and a heart.

Answer to Problem 24CQ

The probability of getting a 5 and a heart is $\frac{1}{52}$.

Explanation of Solution

Calculation:

Let event D denote getting a 5 and a heart.

The possible number of outcomes for getting a 5 and a heart is 1.

That is, there are 1 outcomes for getting an event D.

The formula for probability of event D is,

$P\left(D\right)=\frac{\text{Number of outcomes in }D}{\text{Total number of outcomes in the sample space}}$

Substitute 1 for ‘Number of outcomes in D’ and 52 for ‘Total number of outcomes in the sample space’,

$P\left(D\right)=\frac{1}{\text{52}}$

To determine

To obtain: The probability of getting a king.

Answer to Problem 24CQ

The probability of getting a king is $\frac{1}{13}$.

Explanation of Solution

Calculation:

Let event E denote getting a king.

The possible number of outcomes for getting a king is 4.

That is, there are 4 outcomes for getting an event E.

The formula for probability of event E is,

$P\left(E\right)=\frac{\text{Number of outcomes in }E}{\text{Total number of outcomes in the sample space}}$

Substitute 4 for ‘Number of outcomes in E’ and 52 for ‘Total number of outcomes in the sample space’,

$\begin{array}{c}P\left(E\right)=\frac{4}{\text{52}}\\ =\frac{1}{13}\end{array}$

Thus, probability of getting a king is $\frac{1}{13}$.

To determine

To obtain: The probability of getting a red card.

Answer to Problem 24CQ

The probability of getting a red card is $\frac{1}{2}$.

Explanation of Solution

Calculation:

Let event F denote getting a red card.

The possible number of outcomes for getting a red card is 26.

That is, there are 26 outcomes for getting an event F.

The formula for probability of event F is,

$P\left(F\right)=\frac{\text{Number of outcomes in }F}{\text{Total number of outcomes in the sample space}}$

Substitute 26 for ‘Number of outcomes in E’ and 52 for ‘Total number of outcomes in the sample space’,

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

Thus, probability of getting a red card is $\frac{1}{2}$