Question

The Neckware association of America reported that 3% of ties sold in the United States are bow ties. If 4 customers who purchased a tie randomly selected,find the probability that at least 1 purchased a bow tie

Answer 1

`Pr(at\text{ least 1 purchased a bow tie) = 0.1147}`

Step-by-step explanation:

Pr (people with bow ties) = 3% = 0.03

p = 0.03

Pr (people without bow tie) = 1 - 0.03 = 0.97

q = 0.97

n = 4 customers

`Pr(at\text{ least 1 purchased a bow tie) = 1 - Pr(none purchased a bow tie)}`

To find the probability that at least 1 purchased a bow tie, we will use a binomial probability formula:

`p(x=^{}X)=^nC_xp^xq^{n\text{ - x}}`
`\begin{gathered} \text{Pr(none purchased a bow tie) = p(x = 0)} \\ \text{p(x = 0) = }^4C_0* p^0* q^{4\text{ - }0} \\ \text{p(x = 0) = 1 }*\text{ 1}* q^4=(0.97)^4 \\ \text{p(x = 0) = }0.8853 \\ \\ \text{Pr(none purchased a bow tie) = }0.8853 \end{gathered}`
`\begin{gathered} Pr(at\text{ least 1 purchased a bow tie) = 1 - 0.8853} \\ Pr(at\text{ least 1 purchased a bow tie) = 0.1147} \end{gathered}`