Question

A recent study conducted by a health statistics center found that 27% of households in a certain country had no landline service. This raised concerns about the accuracy of certain surveys, as they depend on random-digit dialing to households via landlines. Pick five households from this country at random. What is the probability that at least one of them does not have a landline _________

Answer 1

We are going to use Binomial Probability Distribution

Probability that they have no landline = q = 27/100 = 0.27

Probability that they have landline = p = 1 - 0.27 = 0.73

Now, to find the probability that at least one of them does not have a landline, we have to find the probability that all the five have a landline first.

So let's find the probability that all the five have a landline:

`\begin{gathered} P(X=x)=^nC_xp^xq^(n-x) \\ ^5C_5(0.73)^5(0.27)^(5-5) \\ P(X\text{ = 5) = }0.2073 \end{gathered}`

So the probability that all the five have a landline = 20.73%

Now is the time to find the probability that at least one of them does not have a landline:

P(at least one has no landline) = 1 - P(All have landline)

= 1 - 0.2073

= 0.7927

So the probability that at least one of them does not have a landline = 79.27%

That's all Please