If 40 of the people in a community used the library in one y

If 40% of the people in a community used the library in one year, find the following probabilities for a random sample of 10 people. Show clear and detailed work. Write correct probability notation. Round the answer to 4 decimal places.

Exactly 4 used the library.

At most 5 used the library.

More than 7 used the library.

If 40% of the people in a community used the library in one year, find the following probabilities for a random sample of 10 people. Show clear and detailed work. Write correct probability notation. Round the answer to 4 decimal places. Exactly 4 used the library. At most 5 used the library. More than 7 used the library.

Solution

Exactly 4 used the library.

Given X follows Binomial distribution with n=10 and p=0.4

P(X=x)=10Cx*(0.4^x)*(0.6^(10-x)) for x=0,1,2,...,10

So P(X=4) =10C4*(0.4^4)*(0.6^(10-4)) = 0.2508227

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At most 5 used the library.

P(X<=5) = P(X=0)+P(X=1)+...+P(X=5)

=10C0*(0.4^0)*(0.6^(10-0))+...+10C5*(0.4^5)*(0.6^(10-5))

=0.8337614

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More than 7 used the library.

P(X>7) = P(X=8)+P(X=9)+P(X=10)

=10C8*(0.4^8)*(0.6^(10-8))+...+10C10*(0.4^10)*(0.6^(10-10))

=0.01229455