Question

A spider climbing out of a well is affected by the weather. When it rains, he falls back down the well with a probability of 1/10. In dry weather, he only falls back down with probability of 1/25. The probability of rain is 1/5.

(i) Draw the tree diagram of these events.

(ii) Find the probability he falls back down the well.

(iii) Find the probability that given he falls it was a rainy day.

Answer 1

(ii)Total probability formula:

$P(falls)=P(falls|rain)P(rain)+P(falls|dry)P(dry)=\frac{1}{10}\cdot\frac{1}{5}+\frac{1}{25}\cdot\frac{1}{25}=0.0216$

(iii)Bayes formula:

$P\left( rain|falls \right) =\frac{P\left( falls|rain \right) P\left( rain \right)}{P\left( falls \right)}=\frac{\frac{1}{10}\cdot \frac{1}{5}}{0.0216}=0.925926$