Question

a researcher randomly selects 95 high school students and asks their handwriting preference and how well they dance. the two - way table displays the data. suppose one of the students is randomly selected. let b = the student prefers writing in block letters and f = the students dancing skills are fair. which of the following is the correct value and interpretation of p(b|f)? p(b|f) = 0.37; given that the student has fair dancing skills, there is a 0.37 probability that they prefer to write with block letters. p(b|f) = 0.37; given that the student prefers to write with block letters, there is a 0.37 probability that they have fair dancing skills. p(b|f) = 0.45; given that the student has fair dancing skills, there is a 0.45 probability that they prefer to write with block letters. p(b|f) = 0.45; given that the student prefers to write with block letters, there is a 0.45 probability that they have fair dancing skills.

Explanation:

Step1: Recall the formula for conditional probability

The formula for conditional probability is \(P(B|F)=\frac{n(B\cap F)}{n(F)}\). Here, \(n(B\cap F)\) is the number of students who prefer block letters and have fair - dancing skills, and \(n(F)\) is the number of students with fair dancing skills.

Step2: Identify the values from the table

From the table, \(n(B\cap F) = 17\) (the cell where "Block" row and "Fair" column intersect) and \(n(F)=38\) (the total of the "Fair" column).

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

$$P(B|F)=\frac{17}{38}\approx0.45$$

The interpretation of \(P(B|F)\): Given that the event \(F\) (student has fair dancing skills) has occurred, we want to find the probability of event \(B\) (student prefers block letters).

Answer:

\(P(B|F) = 0.45\); given that the student has fair dancing skills, there is a \(0.45\) probability that they prefer to write with block letters. So the correct option is \(P(B|F)=0.45\); given that the student has fair dancing skills, there is a \(0.45\) probability that they prefer to write with block letters.