In a large university 20 of the students are business majors

In a large university, 20% of the students are business majors. A random sample of 100 students is selected and their majors are recorded?

What is the probability the sample contains at least 12 business majors?

What is the probability the sample contains less than 15 business majors?

What is the probability the sample contains between 12 and 14 business majors?

Please show all steps

Given X follows Binomial distribution with n=100 and p=0.2

P(X=x)=100Cx*(0.2^x)*(0.8^(100-x)) for x=0,1,2,...,100

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What is the probability the sample contains at least 12 business majors?

P(X>=12) = P(X=12)+P(X=13)+...+P(X=100)

=100C12*(0.2^12)*(0.8^(100-12))+....+100C100*(0.2^100)*(0.8^(100-100))

=0.9874251

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What is the probability the sample contains less than 15 business majors?

P(X<15) = P(X=0)+P(X=1)+...+P(X=14)

=100C0*(0.2^0)*(0.8^(100-0))+...+100C14*(0.2^14)*(0.8^(100-14))

=0.08044372

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What is the probability the sample contains between 12 and 14 business majors?

P(12<=X<=14) = P(X=12)+ P(X=13)+P(X=14)

=100C12*(0.2^12)*(0.8^(100-12))+...+100C14*(0.2^14)*(0.8^(100-14))

=0.06786884