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 0.5. You are 90 % 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 find the probability that the plant died because the neighbor forgot to water it, we can use Bayes' theorem.
Let's define the following events:
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 is dead.
According to Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
We know that:
P(B|A) = 0.8 (the plant died if the neighbor forgot to water it)
P(A) = 0.1 (the probability that the neighbor forgot to water the plant)
P(B) = ? (the probability that the plant died)
To find P(B), we need to consider all possible ways the plant could have died: either the neighbor forgot to water it or the plant died despite being watered.
Let's define a new event:
C: The neighbor watered the plant.
We can express P(B) as the sum of the probabilities of the plant dying in each of these two cases:
P(B) = P(B|A) * P(A) + P(B|~A) * P(~A)
P(B|~A) represents the probability of the plant dying despite being watered, which is given as 0.5.
P(~A) is the complement of event A, i.e., the probability that the neighbor remembered to water the plant, which is 1 - P(A) = 1 - 0.1 = 0.9.
Substituting the values, we have:
P(B) = 0.8 * 0.1 + 0.5 * 0.9
Now, we can calculate P(A|B):
P(A|B) = (0.8 * 0.1) / (0.8 * 0.1 + 0.5 * 0.9)
Simplifying the expression, we find:
P(A|B) = 0.08 / (0.08 + 0.45)
Calculating the numerator and denominator separately:
P(A|B) = 0.08 / 0.53
Finally, we can determine the probability that the neighbor forgot to water the plant given that the plant is dead:
P(A|B) ≈ 0.15 or 15%
Therefore, the probability that the plant died because the neighbor forgot to water it is approximately 15%.