Question

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?
no, p(male) does not equal p(male | geometry)
no, p(male) equals p(male | geometry)
yes, p(male) does not equal p(male | geometry)
yes, p(male) equals p(male | geometry)

Explanation:

Step1: Calculate the probability that a randomly - chosen tenth - grader is male

The total number of tenth - graders is \(34 + 33+40 + 47=154\).
The number of male students is \(33 + 47 = 80\).
The probability \(P(\text{male})=\frac{\text{Number of male students}}{\text{Total number of students}}=\frac{80}{154}=\frac{40}{77}\approx0.519\).

Step2: Calculate the probability that a randomly - chosen tenth - grader is male given that the tenth - grader prefers geometry

The number of students who prefer geometry is \(40 + 47=87\).
The number of male students who prefer geometry is \(47\).
By the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), in the case of frequency - based probability \(P(\text{male}|\text{geometry})=\frac{\text{Number of male students who prefer geometry}}{\text{Number of students who prefer geometry}}=\frac{47}{87}\approx0.540\).

Step3: Check for independence

Two events \(A\) (male) and \(B\) (geometry) are independent if \(P(A)=P(A|B)\).
Since \(\frac{40}{77}\approx0.519
eq\frac{47}{87}\approx0.540\), the events are not independent.

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 \(\frac{47}{87}\).
The answer for the independence question is: no, \(P(\text{male})\) does not equal \(P(\text{male}|\text{geometry})\)