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 0.5. 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?

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 0.5. 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?

Question
Probability
asked 2021-03-05
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 0.5. 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?

Answers (1)

2021-03-06
Let A={plant died},and H1={neighbour watered the plant}, H2={neighbour did not water the plant} PSK We need to find \(\displaystyle{P}{\left({H}{2}{\mid}{A}\right)}\). We will use the Bayes' Theorem (extended form): PSK \(\displaystyle{P}{\left({H}{2}{\mid}{A}\right)}={P}{\left({H}{2}\right)}{P}\frac{{{A}{\mid}{H}{2}}}{{{P}{\left({H}{1}\right)}{P}{\left({A}{\mid}{H}{1}\right)}+{P}{\left({H}{2}\right)}{P}{\left({A}{\mid}{H}{2}\right)}}}=\frac{{{0.1}\cdot{0.85}}}{{{\left({0.9}\cdot{0.5}\right)}+{\left({0.1}\cdot{0.85}\right)}}}=\frac{{0.085}}{{0.535}}\) PSK Therefore, PSK \(\displaystyle{P}{\left({H}{2}{\mid}{A}\right)}=\frac{{17}}{{107}}\)
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