Chapter 5.3, Problem 67E

To determine

Probabilities are to be ranked in ascending order.

Answer to Problem 67E

From smallest to largest,

  $P\left(\text{B}\right)<P\left(\text{B}|\text{T}\right)<P\left(\text{T}\right)<P\left(\text{T}|\text{B}\right)$

Explanation of Solution

Given information:

Events:

T: Person is over 6 feet tall

B: Person is a professional basketball player

Probabilities:

  $P\left(\text{T}\right)$ : Probability for person over 6 feet tall

  $P\left(\text{B}\right)$ : Probability for professional basketball player

  $P\left(\text{B}|\text{T}\right)$ : Conditional probability for over 6 feet tall professional basketball player

  $P\left(\text{T}|\text{B}\right)$ : Conditional probability for professional basketball player over 6 feet tall

According to the universal fact,

Most basketball players are very tall.

Thus,

  $P\left(\text{T}|\text{B}\right)$ will be the largest probability.

Since, there are a lot of tall people.

Thus,

  $P\left(\text{T}\right)$ should be the next largest probability.

Since, most tall people do not play basketball.

Thus,

  $P\left(\text{B}|\text{T}\right)$ is smaller than

  $P\left(\text{T}\right)$ .

Since, tall people are more likely to play basketball.

Thus,

  $P\left(\text{B}|\text{T}\right)$ is larger than $P\left(\text{B}\right)$ .

Thereby,

Arranging the probabilities from smallest to largest:

  $P\left(\text{B}\right)<P\left(\text{B}|\text{T}\right)<P\left(\text{T}\right)<P\left(\text{T}|\text{B}\right)$