Question

a researcher randomly selects 95 high school swimmers and asks them which swim stroke is their strongest and which bathing suit brand they prefer, brand a or brand b. the two - way table displays the data. suppose one of the students is randomly selected. let ( b = ) the student prefers brand b and ( f = ) the students strongest stroke is freestyle. which of the following is the correct value and interpretation of ( p(f|b) )?
( p(f|b)=0.51 ); given that the student prefers brand b, there is a 0.51 probability that their strongest stroke is freestyle.
( p(f|b)=0.51 ); given that the students strongest stroke is freestyle, there is a 0.51 probability that they prefer brand b.
( p(f|b)=0.58 ); given that the student prefers brand b, there is a 0.57 probability that their strongest stroke is freestyle.
( p(f|b)=0.58 ); given that the students strongest stroke is freestyle, there is a 0.57 probability that they prefer brand b.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(F|B)=\frac{P(F\cap B)}{P(B)}\). In terms of counts from the two - way table, \(P(F|B)=\frac{\text{Number of students who prefer brand B and have freestyle as strongest stroke}}{\text{Number of students who prefer brand B}}\).

Step2: Identify the counts from the table

From the table, the number of students who prefer brand B and have freestyle as their strongest stroke is \(n(F\cap B) = 26\), and the number of students who prefer brand B is \(n(B)=51\).

Step3: Calculate \(P(F|B)\)

$$P(F|B)=\frac{26}{51}\approx 0.51$$

The interpretation of \(P(F|B)\) is the probability that a student's strongest stroke is freestyle given that the student prefers brand B.

Answer:

\(P(F|B) = 0.51\); given that the student prefers brand B, there is a \(0.51\) probability that their strongest stroke is freestyle.