Chapter 0.5, Problem 4E

(a)

To determine

Check whether the events are mutually exclusive or not mutually exclusive. Also, find the probability.

Answer to Problem 4E

The probability of freshmen or female is $\frac{13}{20}$ .

Explanation of Solution

Given Information:

In French class, 10 freshmen, 8 sophomores, 2 juniors

Female students = 9 freshmen, 2 sophomores, 1 junior

Formula Used:

Two events are mutually exclusive, when they occur at same time.

Calculation:

Find the total members in French class.

Total members in French class $\begin{array}{c}=10+8+2\\ =20\end{array}$

Consider the probability $P$ (freshmen or female).

The event is not mutually exclusive because there are 9 freshmen as female.

Now, find the probability of freshmen or female.

  $\begin{array}{c}P\left(\text{freshmen}\text{or}\text{female}\right)=\frac{P\left(\text{freshmen}\right)}{\text{Total}\text{outcomes}}+\frac{P\left(\text{female}\right)}{\text{Total}\text{outcomes}}-\frac{P\left(\text{freshmen}\text{and}\text{female}\right)}{\text{Total}\text{outcomes}}\\ =\frac{10}{20}+\frac{12}{20}-\frac{9}{20}\\ =\frac{10+12-9}{20}\\ =\frac{13}{20}\end{array}$

Hence, the probability of freshmen or female is $\frac{13}{20}$ .

(b)

To determine

Check whether the events are mutually exclusive or not mutually exclusive. Also, find the probability.

Answer to Problem 4E

The probability of sophomore or male is $\frac{1}{2}$ .

Explanation of Solution

Given Information:

In French class, 10 freshmen, 8 sophomores, 2 juniors

Female students = 9 freshmen, 2 sophomores, 1 junior

Formula Used:

Two events are mutually exclusive, when they occur at same time.

Calculation:

Find the total members in French class.

Total members in French class $\begin{array}{c}=10+8+2\\ =20\end{array}$

Consider the probability $P$ (sophomoreor or male)

  $\begin{array}{c}Malesophomores=Totalsophomores–Femalesophomores\\ =8-2\\ =6\end{array}$

The event is not mutually exclusive because there are 6 sophomores as male.

Now, find the probability of sophomore or male.

  $\begin{array}{c}P\left(\text{sophomore or male}\right)=\frac{P\left(\text{sophomore}\right)}{\text{Total}\text{outcomes}}+\frac{P\left(\text{male}\right)}{\text{Total}\text{outcomes}}-\frac{P\left(\text{sophomore}\text{and}\text{male}\right)}{\text{Total}\text{outcomes}}\\ =\frac{8}{20}+\frac{8}{20}-\frac{6}{20}\\ =\frac{8+8-6}{20}\\ =\frac{10}{20}\\ =\frac{1}{2}\end{array}$

Hence, the probability of sophomore or male is $\frac{1}{2}$ .

(c)

To determine

Check whether the events are mutually exclusive or not mutually exclusive. Also, find the probability.

Answer to Problem 4E

The probability of freshman or sophomore is $\frac{9}{10}$ .

Explanation of Solution

Given Information:

In French class, 10 freshmen, 8 sophomores, 2 juniors

Female students = 9 freshmen, 2 sophomores, 1 junior

Formula Used:

Two events are mutually exclusive, when they occur at same time.

Calculation:

Find the total members in French class.

Total members in French class $\begin{array}{c}=10+8+2\\ =20\end{array}$

Consider the probability $P$ (freshman or sophomore).

The event is mutually exclusive because a student cannot be both freshman or sophomore.

Now, find the probability of freshman or sophomore.

  $\begin{array}{c}P\left(\text{freshman or sophomore}\right)=\frac{P\left(\text{freshman}\right)}{\text{Total}\text{outcomes}}+\frac{P\left(\text{sophomore}\right)}{\text{Total}\text{outcomes}}\\ =\frac{10}{20}+\frac{8}{20}\\ =\frac{10+8}{20}\\ =\frac{18}{20}\\ =\frac{9}{10}\end{array}$

Hence, the probability of freshman or sophomore is $\frac{9}{10}$ .