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.
(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"
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