You ask your neighbor to water a sickly plant while you are on vacation. Without water, it will die with probability .8; with water, it will die with probability .15. You are 90 percent certain that your neighbor will remember to water the plant. (a) What is the probability that the plant will be alive when you return? (b) If the plant is dead upon your return, what is the probability that your neighbor forgot to water it?

Answer:(a) The probability is 0.785(b) The probability is 0.3721Step-by-step explanation:Let's call A the event that the plant is alive, D the event that the plant die, W the event that your neighbor remember to water the plant and NW the event that your neighbor doesn't remember to water the plant.The probability P(A) that the plant will be alive when you return is calculate as the sum of:P(A) = P(W∩A) + P(NW∩A)So, the probability P(W∩A) that your neighbor water the plant and it is alive is calculate as:P(W∩A) = 0.9 * 0.85 = 0.765Where 0.9 is the probability that your neighbor remember to water the plant and 0.85 is equal to (1-0.15) and it is the probability that the plant is alive given that the plant was water.At the same way, the probability P(NW∩A) that the neighbor didn't water the plant and it is alive is calculate as:P(NW∩A) = 0.1 * 0.2 = 0.02Then, P(A) is:P(A) = 0.765 + 0.02 = 0.785On the other hand, the probability P(NW/D) that your neighbor forgot to water the plant given that is dead is:P(NW/D) = P(NW∩D)/P(D)Where P(D) = P(W∩D) + P(NW∩D)So, the probability P(W∩D) that the neighbor water the plant and it dies and  the probability P(NW∩D) that the neighbor didn't water the plant and it dies are calculate as:P(W∩D) = 0.9 * 0.15 = 0.135P(NW∩D) = 0.1 * 0.8 = 0.08Then, P(D) is:P(D) = 0.135 + 0.08 = 0.215Finally, P(NW/D) is:P(NW/D) = 0.08/0.215 = 0.3721