Question

Fred Friendly, CPA, has 20 tax returns to prepare before the April 15th deadline. It is late at night so he decides to do two more before going home. In his stack of accounts, 12 are personal, 5 are businesses, and 3 are for charitable organizations. If he selects the two returns at random, what is the probability:
a. Both are businesses?
b. At least one is a business?

Answer 1

Combination is the selection of r objects out of n different objects and order of the selection of objects does not matter. If there are 3 objects A, B and C and we have to select 2 objects out of three. Then selection AB and BA are same. Total number of combinations of r objects out of n objects is denoted by nCr and given by

 

nCr = n!/r!(n – r)!

 

Total number of tax returns is,

12 + 5 + 3 = 20

 

Number of ways of selecting 2 returns out of 20 is,

20C2 = 20!/2!(20 – 2)!

      = 190

(a)

Out of 20 tax returns, 5 are businesses so number of ways of selecting 2 taxes out of 5 is,

5C2 = 5!/2!(5 – 2)!

       = 10

 

The probability of getting both returns of businesses is:

P(both business) = 10/190

                               = 0.0526

 

Hence, the required probability is 0.0526.

 

(b)

Out of 20 tax returns, 5 are businesses so number of ways of selecting 2 taxes out of reaming 15 is,

15C2 = 15!/2!(15 – 2)!

           = 105

T

he probability of getting none returns of businesses is,

P(none business) = 105/190

                           = 0.5526

 

The probability of getting at least one return of businesses is:

1 – 0.5226 = 0.04474

 

Hence, the required probability is 0.4474.

(a) 10

(b) 105

Hence, the required probability is 0.4474.