Question

A recent survey found that 30% of telephone users have switched completely to cell phone use (i.e. they do not have landlines in their homes). A random sample of 10 of these customers is selected. What is the probability that exactly 30% of these 10 telephone users do not have landlines in their homes

Answer 1

The probability that exactly 30% of these 10 telephone users do not have landlines in their homes is 0.2668.

Explanation:

We are given that a recent survey found that 30% of telephone users have switched completely to cell phone use (i.e. they do not have landlines in their homes).

A random sample of 10 of these customers is selected.

The above situation can be represented through binomial distribution;

`P(X = r) = \binom{n}{r}* p^(r) * (1-p)^(n-r); x = 0,1,2,3,......`

where, n = number of trials (samples) taken = 10 customers

r = number of success = 30% of 10 = 3

p = probability of success which in our question is the probability

that telephone users do not have landlines in their homes,

i.e. p = 30%

Let X = Number of telephone users who do not have landlines in their homes

So, X ~ Binom(n = 10, p = 0.30)

Now, the probability that exactly 30% of these 10 telephone users do not have landlines in their homes is given by = P(X = 3)

P(X = 3) =
`\binom{10}{3}* 0.30^(3) * (1-0.30)^(10-3)`

=
`120 * 0.30^(3) * 0.70^(7)`

= 0.2668