You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.7. With water it will die with probability 0.6. You are 84 % certain 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?
To determine the probability that the plant died because the neighbor forgot to water it, we can use Bayes' Theorem. Let's define the events as follows:
A: The plant died because the neighbor forgot to water it.
B: The plant died.
We want to find P(A|B), which represents the probability that the neighbor forgot to water the plant given that the plant has died.
According to Bayes' Theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
We know that P(B|A) is the probability that the plant died given that the neighbor forgot to water it. This is given as 0.7.
P(A) is the probability that the neighbor forgot to water the plant, which is based on the certainty level given. Since we are 84% certain that the neighbor will remember to water the plant, the probability that the neighbor forgot to water it is 1 - 0.84 = 0.16.
P(B) is the overall probability that the plant has died, regardless of the neighbor's actions. We need to consider two cases: the neighbor forgot to water the plant and the neighbor remembered to water the plant.
P(B) = P(B|A) * P(A) + P(B|~A) * P(~A)
P(B|~A) is the probability that the plant died given that the neighbor remembered to water it, which is given as 0.6.
P(~A) is the complement of P(A) and represents the probability that the neighbor remembered to water the plant.
P(~A) = 1 - P(A) = 1 - 0.16 = 0.84
Now we can calculate P(B):
P(B) = P(B|A) * P(A) + P(B|~A) * P(~A)
= 0.7 * 0.16 + 0.6 * 0.84
= 0.112 + 0.504
= 0.616
Finally, we can substitute these values into Bayes' Theorem to find P(A|B):
P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.7 * 0.16) / 0.616
= 0.112 / 0.616
≈ 0.182
Therefore, the probability that the plant died because the neighbor forgot to water it is approximately 0.182, or 18.2%.