Question

Consider the chance experiment in which both tennis racket head size and grip size are noted for a randomly selected customer at a particular store. The six possible outcomes (simple events) and their probabilities are displayed in the following table: a. The probability that grip size is \(4 \frac{1}{2}\) in. (event \(\mathrm{A}\) ) is $$ P(A)=P\left(O_{2} \text { or } O_{5}\right)=.20+.15=.35 $$ How would you interpret this probability? b. Use the result of Part (a) to calculate the probability that grip size is not \(4 \frac{1}{2}\) in. c. What is the probability that the racket purchased has an oversize head (event \(B\) ), and how would you interpret this probability? d. What is the probability that grip size is at least \(4 \frac{1}{2}\) in.?

Short answer

The probabilities are as follows: a) There is a 35% chance that a randomly selected customer will choose a grip size of \(4 \frac{1}{2}\) in. b) There's a 65% chance a customer will choose a racket with a grip size not equal to \(4 \frac{1}{2}\) in. c) The probability \(P(B)\) is the chance that a customer will choose an oversize head racket. d) The probability \(P(C)\) is the chance a customer will choose a racket with a grip size of \(4 \frac{1}{2}\) in. or more. Exact numerical values for \(P(B)\) and \(P(C)\) depend on the missing values in the question.

Step-by-step solution

Interpret the Probability of Event A

The question asks to interpret \(P(A)=.35\). This stands for 'The probability that grip size is \(4 \frac{1}{2}\) inches is 0.35'. Interpretation: There is a 35% chance that a randomly selected customer at the store will choose a tennis racket with a grip size of \(4 \frac{1}{2}\) inches.

Calculate the Probability of the complement of Event A

The complement of an event A (denoted \(A'\)) consists of all outcomes not in A. So, to calculate the probability that grip size is not \(4 \frac{1}{2}\) inches, we subtract \(P(A)\) from 1: \(P(A') = 1 - P(A) = 1 - 0.35 = 0.65\). So there's a 65% chance a randomly selected customer will choose a racket with a grip size not equal to \(4 \frac{1}{2}\) inches.

Determine the Probability of Event B

The problem might give the probabilities for event B. Then, it's just a matter of summing those probabilities. Similar to Step 1, add up the probabilities of simple outcomes that result in an oversize head. Let's represent this probability as \(P(B)\). The interpretation is: there's a \(P(B)\) chance that a randomly selected customer at the store will choose a racket with an oversize head.

Calculate the Probability that Grip Size is at least \(4 \frac{1}{2}\) inches

Find all outcomes where the grip size is \(4 \frac{1}{2}\) inches or more and add their probabilities. Let's denote this probability by \(P(C)\). So, \(P(C)\) is the chance a randomly selected customer will choose a racket with a grip size of \(4 \frac{1}{2}\) in. or more.