Question with conditional probability

Question

You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.9. With water it will die with probability 0.5. You are 87 % certain the neighbor will remember to water the plant.

When you are on vacation, find the probability that the plant will die. Ok so for this I was thinking you just multiply.9 x .5 to get .45 from an example I have seen. Why would this be wrong?

You come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it?

This is what I have done and it keeps coming back wrong. P(neighbor forgets|plant dies w/o water) = (1-.87)(.9) = .117

What do I keep doing wrong?

Answer 1

HINT:

The plant dying while you're on vacation happens if either:

1. The neighbor waters it but it dies
2. The neighbors forgets to water and then it dies.

The probabilities for each are different, and they need to be added together.

Edit: For the second bit, recall the formula $$P(A | B) = \frac{P(A \cap B)}{P(B)}$$ So we need to find P(neighbor forgets | plant dies) = P(neighbor forgets and plant dies) / P(plant dies). Can you complete it now?