You ask a neighbor to water a sickly plant while you are on vacation. Without water the plant will die with probability 0.85. With water it will die with probability 0.55. You are 89 % 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 events:
A: The neighbor forgot to water the plant
B: The plant is dead
We know the following probabilities:
P(A) = 1 - 0.89 = 0.11 (probability that the neighbor forgot to water the plant)
P(B|A) = 0.85 (probability that the plant dies if the neighbor forgets to water it)
P(B|A') = 0.55 (probability that the plant dies if the neighbor waters it)
We want to find P(A|B), the probability that the neighbor forgot to water the plant given that it is dead. Bayes' theorem states:
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(B) = 0.85 * 0.11 + 0.55 * 0.89
P(B) = 0.0935 + 0.4895
P(B) = 0.583
Now we can substitute the values into Bayes' theorem:
P(A|B) = (0.85 * 0.11) / 0.583
P(A|B) ≈ 0.1606
Therefore, the probability that the neighbor forgot to water the plant given that it is dead is approximately 0.1606, or 16.06%.