The chances of a tax return being audited are about18 in 1,000 if an income is l

Question

The chances of a tax return being audited are about18 in 1,000 if an income is less than $100,000 and 35 in 1,000 if an income is $100,000 or more. Complete parts a through e. a.

What is the probability that a taxpayer with income less than $100,000 will be audited? With income of $100,000 or more?

P(taxpayer with income less than $100,000 is audited)equals= ? (Type an integer or a decimal.)

What is the probability that a taxpayer with income of $100,000 or more will be audited? P(taxpayer with income of $100,000 or higher is audited)=?

(Type an integer or a decimal.)

b. If Three taxpayers with incomes under $100,000 are randomly selected, what is the probability that exactly one will be audited? That more than one will be audited? P(x=1) = ? (Round to four decimal places as needed.) What is the probability that more than one will be audited? P(x>1)equals= nothing (Round to four decimal places as needed.)

c. Repeat part b assuming that fivefive taxpayers with incomes of $100,000 or more are randomly selectedP(x=1)= ? (Round to four decimal places as needed.) What is the probability that more than one will be audited? P(x>1) = nothing (Round to four decimal places as needed.)

d. If two taxpayers with incomes under $100,000 are randomly selected and two with incomes more than $100,000 are randomly selected, what is the probability that none of these taxpayers will be audited? P(none of the taxpayers will be audited)= ?(Round to four decimal places as needed.)

e What assumptions did you have to make in order to answer these questions?

A. We must assume that the variables are binomial random variables. We must assume that the trials are identical, the probability of success is the same from trial to trial, and that the trials are independent.

B. We must assume that the variables are binomial random variables. We must assume that the trials are identical, the probability of success varies from trial to trial, and that the trials are dependent.

C. We must assume that the variables are random variables. We must assume that the trials are identical, and the probability of success varies from trial to trial. D. We must assume that the variables are binomial random variables. We must assume that the trials are identical and dependent.

D. We must assume that the variables are random variables. We must assume that the trials are identical, and the probability of success varies from trial to trial.

Explanation / Answer

(a)
Probability(taxpayer with income less than 100000 is audited) = 18/1000 = 0.018

Probability(taxpayer with income equals to or more than 100000 is audited) = 35/1000 = 0.035

(b)
p = 0.018

If 3 taxpayer under $100000 are selected at random

P(x = 1) = 3C1*(0.018)*(1-0.018)^2 = 0.05207

P(x = 0) = (1 - 0.018)^3 = 0.947

P(x > 1) = 1 - P(x = 1) - P(x = 0)
= 1 - 0.05207 - 0.947
= 0.00093

(c)
p = 0.018

If 5 taxpayer under $100000 are selected at random

P(x = 1) = 5C1*(0.018)*(1-0.018)^4 = 0.0837

P(x = 0) = (1 - 0.018)^5 = 0.9132

P(x > 1) = 1 - P(x = 1) - P(x = 0)
= 1 - 0.0837 - 0.9132
= 0.0031

(d)
Required probability = (1-0.018)^2 * (1-0.035)^2 = 0.898

e)
option (A) is correct

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