Without water the plant will die with probability 0.9. with water it will die with probability 0.5. you are 85 % certain the neighbor will remember to water the plant. when you return from vacation, the plant is dead. what is the probability that the plant died because your neighbor forgot to water it?

The probability is 0.2411

Step-by-step explanation:

Let's call: W the event that the neighbor will remember to water the plant, NW th event that the neighbor will not remember to water the plant and D the event that the plant will die.

So, the probability that the plant died because your neighbor forgot to water it is:

P(NW/D) = P(NW∩D)/P(D)

Where P(D) = P(NW∩D) + P(W∩D)

P(NW∩D) is the probability that the neighbor forgot to water the plant and the plant is died. That is calculated as:

P(NW∩D) = 0.15*0.9 = 0.135

Because 0.15 is the probability that the neighbor will not remember to water the plant and 0.9 is the probability that the plant given that it is not water.

At the same way, P(W∩D) is the probability that the neighbor water the plant and the plant is died. That is calculated as:

P(W∩D) = 0.85*0.5 = 0.425

Because 0.85 is the probability that the neighbor will not remember to water the plant and 0.5 is the probability that the plant will die given that it is water.

Then, The probability P(D) is:

P(D) = 0.135+0.425

P(D) = 0.56

Finally The probability P(NW/D) that the plant died because your neighbor forgot to water it is:

P(NW/D) = 0.135/0.56

P(NW/D) = 0.2411