Question

The one-time fling! Have you ever purchased an article of clothing (dress, sports jacket, etc.), worn the item once to a party, and then returned the purchase? This is called a one-time fling. About 15% of all adults deliberately do a one-time fling and feel no guilt about it! In a group of eight adult friends, what is the probability of the following?

a. no one has done a one time fling
b. at least one person has done a one time fling
c. no more than two people have done a one time fling

Answer 1

`(a)\ P(x = 0) = 0.2725`

`(b)\ P(x \ge 1) =0.7275`

`(c)\ P(x \le 2) = 0.8948`

Explanation:

Given

`n = 8` --- 8 friends

`p = 15\%` --- proportion that one-time fling

This question is an illustration of binomial probability, and it is represented as:

`P(X = x) = ^nC_x* p^x * (1 - p)^(n-x)`

Solving (a): P(x = 0) --- None has done one time fling

`P(x = 0) = ^8C_0* (15\%)^0 * (1 - 15\%)^(8-0)`

`P(x = 0) = 1* 1 * (1 - 0.15)^(8)`

`P(x = 0) = 0.85^8`

`P(x = 0) = 0.2725`

Solving (b):
`P(x \ge 1)`

To do this, we make use of compliment rule:

`P(x = 0) + P(x \ge 1) =1`

Rewrite as:

`P(x \ge 1) =1 - P(x = 0)`

`P(x \ge 1) =1 - 0.2725`

`P(x \ge 1) =0.7275`

Solving (c):
`P(x\le 2)`--- Not more than 2 has one time fling

This is calculated as:

`P(x\le 2) = P(x = 0) + P(x =1) + P(x = 2)`

We have:

`P(x = 0) = 0.2725`

`P(x = 1) = ^8C_1* (15\%)^1 * (1 - 15\%)^(8-1)`

`P(x = 1) = 8* (0.15) * (1 - 0.15)^7`

`P(x = 1) = 0.3847`

`P(x = 2) = ^8C_2* (15\%)^2 * (1 - 15\%)^(8-2)`

`P(x = 2) = 28* (0.15)^2 * (1-0.15)^6`

`P(x = 2) = 0.2376`

So:

`P(x\le 2) = P(x = 0) + P(x =1) + P(x = 2)`

`P(x \le 2) = 0.2725 + 0.3847 + 0.2376`

`P(x \le 2) = 0.8948`