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 82 % 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 solve this problem, we can use Bayesian probability. We need to calculate the probability that the neighbor forgot to water the plant given that the plant is dead.
Let's define the events:
A: Neighbor forgot to water the plant
B: The plant is dead
We are given the following probabilities:
P(B|A') = 0.7 (probability of the plant dying if the neighbor did not forget to water it)
P(B|A) = 0.6 (probability of the plant dying if the neighbor forgot to water it)
P(A) = 0.82 (probability that the neighbor remembered to water the plant)
We want to find P(A|B), the probability that the neighbor forgot to water the plant given that the plant is dead.
We can use Bayes' theorem to calculate this:
P(A|B) = (P(B|A) * P(A)) / P(B)
We can calculate the denominator using the law of total probability:
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
We are given P(B|A') = 0.7 and P(A') = 1 - P(A) = 1 - 0.82 = 0.18.
Now we can substitute these values into the equation to calculate P(A|B):
P(A|B) = (0.6 * 0.82) / (0.6 * 0.82 + 0.7 * 0.18)
Simplifying this, we get:
P(A|B) = 0.4928 / (0.4928 + 0.126)
Therefore, the probability that the plant died because the neighbor forgot to water it is approximately 0.796, or 79.6%.