18. Apple sells two models of iPhone 7: the 4.7-inch model and the 5.5-inch Plus

Question

18. Apple sells two models of iPhone 7: the 4.7-inch model and the 5.5-inch Plus model. Since
last September, 75% of the sold iPhone 7’s were the 4.7-inch model and 25% were the 5.5-
inch Plus model. Of those customers who bought the 4.7-inch iPhone 7, 45% purchased an
AppleCare extended warranty; whereas 65% of all 5.5-inch iPhone 7 Plus buyers purchased
an AppleCare extended warranty. What is the probability that the next iPhone 7 buyer will
purchase an AppleCare extended warranty? (Hint: use the law of total probability.)

I know the answer is .50 but can anyone expline why please

Explanation / Answer

P(4.7 inch model) = 0.75

P(5.5 inch model) = 0.25

P(AppleCare extended warranty | 4.7 inch model) = 0.45

P(AppleCare extended warranty | 5.5 inch model) = 0.65

P(AppleCare extended warranty) = P(AppleCare extended warranty | 4.7 inch model) * P(4.7 inch model) + P(AppleCare extended warranty | 5.5 inch model) * P(5.5 inch model)

                                                    = 0.45 * 0.75 + 0.65 * 0.25

                                                    = 0.5 (ans)

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