Problem
A store carries four brands of $C D$ players, J, G, P, and S. From past records, the manager found that the relative frequency of brand choice among customers varied. Which of the following probability assignments for a particular customer choosing a particular brand of $C D$ player would have to be rejected? Why? (A) $P(J)=0.10, P(G)=0.47, P(P)=0.29, P(S)=0.57$ (B) $P(J)=0.17, P(G)=0.45, P(P)=0.24, P(S)=0.14$ (C) $P(J)=0.36, P(G)=-0.12, P(P)=0.23, P(S)=0.53$ Should probability assignment (A) be rejected? Why or why not? A. No, because each probability is between 0 and 1 (inclusive), and the sum of theprobabilites is 1 . B. Yes, because one of the probabilities is negative. C. Yes, because the sum of all the events is not 1. D. Yes, because each probability is between 0 and 1 (inclusive), and the sum of the probabilities is 1 .
Solution
Step 1 ：Assign the given probabilities to each brand: \(P(J) = 0.10\), \(P(G) = 0.47\), \(P(P) = 0.29\), \(P(S) = 0.57\).
Step 2 ：Calculate the total probability: \(P(J) + P(G) + P(P) + P(S) = 1.43\).
Step 3 ：Since the total probability is not equal to 1, the probability assignment (A) should be rejected.
Step 4 ：Final Answer: \(\boxed{\text{(C) Yes, because the sum of all the events is not 1.}}\)
