Chapter 10.1, Problem 31E

Solution

To calculate: The probability she is from Mr. Abel’s class if a West High precalculus student chosen at random happens to be a girl.

The probability that she is from Mr. Abel’s class is $\frac{27}{44}$ .

Given information:

There are two precalculus sections at West High School. Mr. Abel’s class has $12$ girls and $8$ boys, and Mr. Bonitz’s class has $10$ girls and $15$ boys.

Calculation:

The formula used to find conditional probability is:

  $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

In our case the formula used to calculate to probability of choosing someone from Mr. Abel’s, given that they are a girl is:

  $P\left(\text{Mr}\text{. Abel's class}|\text{girl}\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}$

As there are two classes and there is an equal chance of choosing either.

The probability of two events occurring simultaneously is calculated using the formula:

  $P\left(A\text{ and }B\right)=P\left(A\right)\cdot P\left(B\right)$

As there are two classes and there is an equal chance of picking a student from either of them, it follows:

  $\begin{array}{l}P\left(\text{Mr}\text{. Abel's class}\right)=\frac{1}{2}\\ =0.5\end{array}$

Using the total number of students and girls in Mr. Abel's class, it concludes that:

  $P\left(\text{Girl from Mr}\text{. Abel's class}\right)=\frac{12}{20}$

Using the total number of students and girls in both classes, it concludes that:

  $\begin{array}{l}P\left(\text{Girl}\right)=\frac{12+10}{45}\\ =\frac{22}{45}\end{array}$

Substitute the values into the conditional probability:

  $\begin{array}{l}P\left(\text{Mr}\text{. Abel's class}|\text{girl}\right)=\frac{\frac{1}{2}\times \frac{12}{20}}{\frac{22}{45}}\\ =\frac{27}{44}\end{array}$

Hence, the probability that she is from Mr. Abel’s class if a West High precalculus student chosen at random happens to be a girl is $\frac{27}{44}$ .