Question

determining if events are independent
the 154 tenth - graders at wilson high school were polled on whether they enjoyed their algebra or geometry course more. the results are shown below.
algebra: 34 female, 33 male
geometry: 40 female, 47 male
use the drop - down menus to answer the questions.
what is the probability that a randomly chosen tenth - grader is male?
what is the probability that a randomly chosen tenth - grader is male given that the tenth - grader prefers geometry?
are the events \male\ and \geometry\ independent?

Explanation:

Step1: Calculate \( P(\text{male}) \)

Total number of students \( n = 154 \).
Number of male students \(=33 + 47=80\).
\( P(\text{male})=\frac{\text{Number of male students}}{\text{Total number of students}}=\frac{80}{154}=\frac{40}{77}\approx0.519 \)

Step2: Calculate \( P(\text{male}|\text{geometry}) \)

Number of students who prefer geometry \(=40 + 47 = 87\).
Number of male students who prefer geometry \( = 47\).
By the formula \( P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{n(A\cap B)}{n(B)}\), \( P(\text{male}|\text{geometry})=\frac{\text{Number of male - geometry students}}{\text{Number of geometry students}}=\frac{47}{87}\approx0.540 \)

Step3: Check for independence

Two events \( A\) (male) and \( B\) (geometry) are independent if \( P(A)=P(A|B)\)
Since \( \frac{40}{77}
eq\frac{47}{87}\)

Answer:

The probability that a randomly - chosen tenth - grader is male is \(\frac{40}{77}\).
The probability that a randomly - chosen tenth - grader is male given that the tenth - grader prefers geometry is \(\frac{47}{87}\).
Are the events “male” and “geometry” independent? no, \( P(\text{male})\) does not equal \( P(\text{male}|\text{geometry})\)