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

Question

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.Continuing the last question, if the next buyer is known to have purchased an AppleCare extended warranty, how likely is it that he or she has bought the 4.7-inch iPhone 7 model? (Hint: use Bayes’ Theorem.) i know the answer is 0.675 can you please explain how

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

P(4.7 inch model | Apple care extended warranty) = P(AppleCare extended warranty | 4.7 inch model) * P(4.7 inch model) / P(Apple care extended warranty)

             = 0.45 * 0.75 / 0.5

             = 0.675 ( ans)

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