You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.8. With water it will die with probability 0.4. You are 87 % certain the neighbor will remember to water the plant.
You come back from the vacation and the plant is dead. What is the probability the neighbor forgot to water it?
To find the probability that the neighbor forgot to water the plant given that it is dead, we can use Bayes' theorem.
Let's define the following events:
A: Neighbor forgot to water the plant
B: Plant is dead
We are given the following probabilities:
P(B|¬A) = 0.8 (probability that the plant dies given the neighbor did not forget to water it)
P(B|A) = 0.4 (probability that the plant dies given the neighbor forgot to water it)
P(A) = 0.13 (probability that the neighbor forgot to water the plant, calculated as (1 - 0.87))
We want to calculate P(A|B), the probability that the neighbor forgot to water the plant given that it is dead.
Using Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
To calculate P(B), we can use the law of total probability:
P(B) = P(B|A) * P(A) + P(B|¬A) * P(¬A)
P(¬A) = 1 - P(A) = 0.87
Substituting the values into the formula, we get:
P(A|B) = (0.4 * 0.13) / ((0.4 * 0.13) + (0.8 * 0.87))
Simplifying the equation further:
P(A|B) = 0.052 / (0.052 + 0.696)
P(A|B) ≈ 0.0693
Therefore, the probability that the neighbor forgot to water the plant given that it is dead is approximately 0.0693 or 6.93%.