# The annual rainfall (in inches) in a certain region is

The annual rainfall (in inches) in a certain region is normally distributed with $$\displaystyle\mu={40}$$ and $$\displaystyle\sigma={4}$$
What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?

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Michele Tipton
Given that the annual rainfall is normally distributed with mean $$\displaystyle\mu={40}$$ inches and standart deviation $$\displaystyle\sigma={4}$$ inches. Let X be the normal random variable of the annual rainfall.
To find: P
Solution: We will first find the probability of rainfall above 50 inches in one year.
$$\displaystyle{P}{\left[{X}{>}{50}\right]}={1}-{P}{\left[{x}\leq{50}\right]}$$
$$\displaystyle={1}-{P}{\left[{\frac{{{X}-\mu}}{{\sigma}}}\leq{\frac{{{50}-{40}}}{{{4}}}}\right]}$$
$$\displaystyle={1}-{P}{\left[{Z}\leq{\frac{{{5}}}{{{2}}}}\right]}$$
$$\displaystyle={1}-\Phi{\left({\frac{{{5}}}{{{2}}}}\right)}$$
$$\displaystyle{1}-{0.9938}$$
$$\displaystyle={0.0062}$$
P[non-occurence of rainfall above 50 inches]=1-0.0062=0.9938
Assuming independence over the years, we get
P[It will take over 10 years or more before an year with a rainfall above 50 inches]$$\displaystyle={\left({0.9938}\right)}^{{{10}}}$$
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