Question

The probability of being audited by IRS goes up as the reported income goes up. It was estimated 25% of the tax filers will be audited in a certain high income group. A group of 12 tax filers in this income group was selected. (Hint: Use Excel BINOM function)

Create a binomial probability table and cumulative probability table reflecting the chances for 0, 1, 2, 3, … 12 of the tax filers in this group to be audited

What is the probability that 6 or more in this group will be audited?

What is the probability that nobody in this group is audited?

What is the probability that all the 12 will be audited?

What is the minimum number of tax payers in this group that IRS should audit so that the cumulative probability of being audited is over .5 for this income group?

Answer 1

below is probability table from excel function binomdist(x,12,0.25,false) where x is data values from 0,1,2...12

x	P(X=x)
0	0.0317
1	0.1267
2	0.2323
3	0.2581
4	0.1936
5	0.1032
6	0.0401
7	0.0115
8	0.0024
9	0.0004
10	0.0000
11	0.0000
12	0.0000

from excel function: binomdist(x,12,0.25,true) below is cumulative  probability table:

x	P(X<=x)
0	0.0317
1	0.1584
2	0.3907
3	0.6488
4	0.8424
5	0.9456
6	0.9857
7	0.9972
8	0.9996
9	1.0000
10	1.0000
11	1.0000
12	1.0000

2)

probability that 6 or more in this group will be audited =P(X>=6)=1-P(X<=5)\

=1-binomdist(5,12,0.25,true)=0.0544

3)

probability that nobody in this group is audited =P(X=0)=binomdist(0,12,0.25,false)=0.0317

4)

probability that all the 12 will be audited=P(X=12)=binomdist(12,12,0.25,false)=5.96*10^-8~ 0.0000

5)

minimum number =critbinom(12,0.25,0.5)=3