A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we denote by $A_n$ the event: ``the customer buys a melon during week $n$''. Thus $p(A_1) = 1$. a. Reproduce and complete the probability tree below, relating to the first three weeks. b. Prove that $p(A_3) = 0{,}85$. c. Given that the customer buys a melon during week 3, what is the probability that he bought one during week 2? Round to the nearest hundredth.
A customer is chosen at random from those who bought a melon during week 1. Among customers who buy a melon in a given week, $90\%$ of them buy a melon the following week; among customers who do not buy a melon in a given week, $60\%$ of them do not buy a melon the following week. For $n \geqslant 1$, we denote by $A_n$ the event: ``the customer buys a melon during week $n$''. Thus $p(A_1) = 1$.\\
a. Reproduce and complete the probability tree below, relating to the first three weeks.\\
b. Prove that $p(A_3) = 0{,}85$.\\
c. Given that the customer buys a melon during week 3, what is the probability that he bought one during week 2? Round to the nearest hundredth.