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 5% of all adults deliberately do a one-time fling and feel no guilt about it! In a group of nine adult friends, what is the probability of the following? (Round your answers to three decimal places.)

(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)=(9C0)(0.05)^0 (1-0.05)^(9-0)=0.630`

b)
`P(X\geq 1)=1-0.630=0.370`

c)
`P(X\leq 2)=0.630+0.299+0.0629=0.992`

Explanation:

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n=9, p=0.05)`

The probability mass function for the Binomial distribution is given as:

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

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Part a

`P(X=0)=(9C0)(0.05)^0 (1-0.05)^(9-0)=0.630`

Part b

`P(X\geq 1)=1-P(X< 1)=1-P(X=0)`

`P(X=0)=(9C0)(0.05)^0 (1-0.05)^(9-0)=0.630`

`P(X\geq 1)=1-0.630=0.370`

Part c

`P(X\leq 2)=P(X=0)+P(X=1)+P(X=2)`

`P(X=0)=(9C0)(0.05)^0 (1-0.05)^(9-0)=0.630`

`P(X=1)=(9C1)(0.05)^1 (1-0.05)^(9-1)=0.299`

`P(X=2)=(9C2)(0.05)^2 (1-0.05)^(9-2)=0.0629`

`P(X\leq 2)=0.630+0.299+0.0629=0.992`