Question

The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are randomly selected from the six (implying 15 equally likely outcomes; if the computers are numbered 1, 2, . . . , 6, then one outcome consists of computers 1 and 2, another consists of computers 1 and 3, and so on).

Required:
a. What is the probability that both selected setups are for laptop computers?
b. What is the probability that both selected setups are desktop machines?
c. What is the probability that at least one selected setup is for a desktop computer?
d. What is the probability that at least on computer of each type is chosen for setup?

Answer 1

a.
`(1)/(15)`

b.
`(2)/(5)`

c.
`(14)/(15)`

d.
`(8)/(15)`

Explanation:

Given that there are two laptop machines and four desktop machines.

On a day, 2 computers to be set up.

To find:

a. probability that both selected setups are for laptop computers?

b. probability that both selected setups are desktop machines?

c. probability that at least one selected setup is for a desktop computer?

d. probability that at least one computer of each type is chosen for setup?

Solution:

Formula for probability of an event E can be observed as:

`P(E) = \frac{\text{Number of favorable cases}}{\text {Total number of cases}}`

a. Favorable cases for Both the laptops to be selected =
`_2C_2` = 1

Total number of cases = 15

Required probability is
`(1)/(15)`.

b. Favorable cases for both the desktop machines selected =
`_4C_2=6`

Total number of cases = 15

Required probability is
`(6)/(15) = (2)/(5)`.

c. At least one desktop:

Two cases:

1. 1 desktop and 1 laptop:

Favorable cases =
`_2C_1* _4C_1 = 8`

2. Both desktop:

Favorable cases =
`_4C_2=6`

Total number of favorable cases = 8 + 6 = 14

Required probability is
`(14)/(15)`.

d. 1 desktop and 1 laptop:

Favorable cases =
`_2C_1* _4C_1 = 8`

Total number of cases = 15

Required probability is
`(8)/(15)`.