The Internal Revenue Service is studying the category of charitable contribution

Question

The Internal Revenue Service is studying the category of charitable contributions. A sample of 34 returns is selected from young couples between the ages of 20 and 35 who had an adjusted gross income of more than $100,000. Of these 34 returns, 8 had charitable contributions of more than $1,000. Suppose 7 of these returns are selected for a comprehensive audit.

What is the probability exactly one of the seven audited had a charitable deduction of more than $1,000? (Round your answer to 4 decimal places.)

What is the probability at least one of the audited returns had a charitable contribution of more than $1,000? (Round your answer to 4 decimal places.)

What is the probability exactly one of the seven audited had a charitable deduction of more than $1,000? (Round your answer to 4 decimal places.)

Explanation / Answer

Based on the data given. Total = 34, Those who have charitable income = 8, and those who dont have charitable income = 34. Therefore,

P(Of having charitable income) = 8/34 = 4/17

P(Of not having charitable income) = 26/34 = 13/17

(b) What is the probability exactly one of the seven audited had a charitable deduction of more than $1,000? (Round your answer to 4 decimal places.)

From the 7 audited, we can choose 1 person in 7C1 ways = 7

Therefore the required probability = 7C1 * (4/17) * (13/17)6 = 0.3293

(c) When we say at least 1,we need probabilities of those having 1contribution+2contributions+3contributions+.....+7contributions. Simply put, we are only excuding the option of 0 people having a probability of charitable contribution.That means the possibility that all 7 do not have a charitable contribution whose probability = (13/17)7.

Therefore,

The required probability is then = 1 - P(That none of the 7 selected have charitable income)..Since sum of all probabilities =1 , i.e P (0Contributions+1 Contribution+2+3+4+5+6+7) = 1

P(At least 1 person audited has a charitable contribution)= 1 - (13/17)7 = 0.8471

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