Cayuga Lake in central New York State is rather deep and wil

Cayuga Lake, in central New York State, is rather deep and will
freeze only after a long period of exceptionally cold weather.

In any given winter, the lake surface either freezes or does not.

Whether the lake freezes in a particular winter is independent of
whether it froze in recent years ... and the probability that lake
will freeze in a particular year is effectively constant = to 5%
based upon empirical evidence.

Using appropriate probability model(s), ... What is the probability

a) ... of the lake freezing exactly once in 10 years?

b) ... of the lake freezing at most once in 10 years?

c) ... of the lake freezing at least five times in 10 years?

d) How many times is the lake expected to freeze in 10 years?

e) What are the reasonable number of times for the lake to
   freeze in 10 years? Justify Your Answer. Explain.

f) What are the reasonable number of times for the lake to
   freeze in 100 years? Justify Your Answer. Explain.

Solution

a) ... of the lake freezing exactly once in 10 years?

Given X follows Binomial distribution with n=10, p=0.05

P(X=x)=10Cx*(0.05^x)*(0.95^(10-x))

So P(X=1)=10C1*(0.05^1)*(0.95^(10-1))=0.3151247

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b) ... of the lake freezing at most once in 10 years?

P(X<=1)=P(X=0)+P(X=1)

=10C0*(0.05^0)*(0.95^(10-0))+10C1*(0.05^1)*(0.95^(10-1))

=0.9138616

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c) ... of the lake freezing at least five times in 10 years?

P(X>=5)=P(X=5)+P(X=6)+...+P(X=10)

=10C5*(0.05^5)*(0.95^(10-5))+...+10C10*(0.05^10)*(0.95^(10-10))

=0.00006

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d) How many times is the lake expected to freeze in 10 years?

expected =n*p=10*0.05=0.5

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e) What are the reasonable number of times for the lake to
   freeze in 10 years? Justify Your Answer. Explain.

0.5 because it is the expected value.

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f) What are the reasonable number of times for the lake to
   freeze in 100 years? Justify Your Answer. Explain.

100*0.05=5 because it is the expected value.