Houseplant death probability

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. Any plant that would die with water would also die without water. There is a 0.81 probability that the neighbor will remember to water the plant.

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

Please justify your answer with an explanation. The best correct answer will be awarded. It will not go to a vote. In the event of no correct answers, the question will be deleted.

P[waters & dies] = 81% * 50% = 40.5%
P[forgets & dies anyway] = 19%* 50% = 9.5%
P[forgets & dies because of it] = 19% * 40% = 7.6% = A
P[dies] = sum of the above = 57.6% = B

P[ dies because forgot | dies] = A/B = 0.1319 <-------

P(water & die) = .81 * .5 = .405
P(water & live) = .81 * .5 = .405
P(forget & die) = .19 * .9 = .171
P(forget & live) = .19 * .1 = .019

P(die) = P(water & die) + P(forget & die) = .405 + .171 = .576

P(forget | die) = P(forget & die) / P(die) = .171 / .576 = .297

Garden variety (pun intended) intro conditional probability.