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)? b = the student prefers writing in block letters and f = the students dancing skills are fair. dancing skill minimal fair good total block 16 17 13 46 handwriting preference curse 9 21 19 49 total 25 38 32 95 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)=0.45; 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 conditional - probability formula

The formula for conditional probability is $P(B|F)=\frac{P(B\cap F)}{P(F)}$. In terms of the two - way table, $P(B|F)=\frac{n(B\cap F)}{n(F)}$, where $n(B\cap F)$ 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 values from the table

From the table, the number of students who prefer block letters and have fair dancing skills $n(B\cap F) = 17$, and the number of students with fair dancing skills $n(F)=38$.

Step3: Calculate the probability

$P(B|F)=\frac{n(B\cap F)}{n(F)}=\frac{17}{38}\approx0.45$. The interpretation of $P(B|F)$ is: given that the student has fair dancing skills, there is a $0.45$ probability that they prefer to write with 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.