Question

a researcher randomly selects 165 vehicles and sees how many miles each car has been driven and the color of the vehicle. the two - way table displays the data. suppose a vehicle is randomly selected. let m = vehicle has been driven many miles and b = selected vehicle is blue. which of the following is the correct value and interpretation of p(b|m)? p(b|m)=0.54; given that the vehicle color is blue, there is a 0.54 probability that it has been driven many miles. p(b|m)=0.36; given that the vehicle has been driven many miles, there is a 0.36 probability that the color is blue. p(b|m)=0.36; given that the vehicle color is blue, there is a 0.36 probability that it has been driven many miles.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(B|M)=\frac{P(B\cap M)}{P(M)}$. In terms of the two - way table, $P(B|M)=\frac{n(B\cap M)}{n(M)}$, where $n(B\cap M)$ is the number of elements in the intersection of $B$ and $M$, and $n(M)$ is the number of elements in $M$.
From the table, the number of vehicles that are blue and have been driven many miles ($n(B\cap M)$) is 34, and the number of vehicles that have been driven many miles ($n(M)$) is $39 + 21+34=94$.

Step2: Calculate the probability

$P(B|M)=\frac{n(B\cap M)}{n(M)}=\frac{34}{94}\approx0.36$.
The interpretation of $P(B|M)$ is: given that the vehicle has been driven many miles, the probability that the color is blue.

Answer:

$P(B|M) = 0.36$; given that the vehicle has been driven many miles, there is a 0.36 probability that the color is blue.