Suppose that the IRS says that 5 of the population cheat on

Suppose that the IRS says that 5% of the population cheat on their income taxes every year and the result is independent of age, ethnicity, and income bracket. lets say this number holds for every year. Suppose that 40% of the population that filed income taxes is between the ages of 18 to 44 years old for the year 2010

a) what is the probability that an adult that filed income taxes for the year 2010 cheated on their income tax and is between the ages of 18 to 44 years old?

b) Suppose we sampled five tax returns at random for the year 2010, What is the probability that at least one person is between the ages of 18 to 44 years old?

c) Suppose we sampled three tax returns at random for the year 2010. What is the probability that all three cheated on their income tax?

Solution

(a) 0.05*0.4 = 0.02

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(b) Given X~Binomial(n=5, p=0.02)

P(X=x)=5Cx*(0.02^x)*(0.98^(5-x))

P(at least one person)

=1-P(no one)

=1-P(X=0)

=1-5C0*(0.02^0)*(0.98^(5-0))

=0.0960792

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(c)0.05^3 =0.000125