Question

Ram’s gardener is not dependable, the probability that he will forget to water the rose bush is $\frac{2}{3}$. The probability of its withering if watered is $\frac{1}{2}$and the probability of withering if not watered is $\frac{3}{4}$. Ram went out of station and upon returning, he finds that the rose bush has withered. If the probability that the gardener did not water the bush is p, then the value of 16p = ___

Answer 1

After coming back home Ram finds rose bush to be withered. It could have happened because the gardener did not water or it could have happened even after the gardener watered. Ram wants to know the probability that the gardener did not water given that the rose bush is withered.
Let us first give names for these events and try to represent the given conditions using variables.
Let W = The event that the bush was withered
R = The event that the gardener remembered to water
R’ = The gardener forgot to water We want to find the probability that the gardener did not water given that the rose bush is withered.
We can represent it as $P\left(\frac{R^{\prime }}{W}\right)$
Using Bayes’ theorem we get $P\left(\frac{R^{\prime }}{W}\right)=\frac{P\left(R^{\prime }\right)P\left(\frac{W}{R^{\prime }}\right)}{P\left(R^{\prime }\right)P\left(\frac{W}{R^{\prime }}\right)+P\left(R\right)P\left(\frac{W}{R}\right)}$
The denominator of this expression is nothing but $P\left(W\right).P\left(\frac{W}{R^{\prime }}\right)$ is the probability of withering given that gardner did not water.
In the question we are given probability of not watering $=P\left(R^{\prime }\right)=\frac{2}{3}$
Probability of watering $=P\left(R\right)=\frac{1}{3}$
Probability that bush withers if watered $=P\left(\frac{W}{R}\right)=\frac{1}{2}$
Probability that bush withers if not watered $=P\left(\frac{W}{R^{\prime }}\right)=\frac{3}{4}$
We have values of all the variables in the equation. Substituting them, we get
$=P\left(\frac{R^{\prime }}{W}\right)=\frac{\frac{2}{3}\times \frac{3}{4}}{\frac{2}{3}\times \frac{3}{4}\times \frac{1}{3}\times \frac{1}{2}}=\frac{3}{4}$
This is given as p. We want to find 16p
$16p=16\times \frac{3}{4}$
=12