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

Answer:

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

Step-by-step 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;

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

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) =  $\binom{10}{3}\times 0.30^{3} \times (1-0.30)^{10-3}$

                         =  $120 \times 0.30^{3} \times 0.70^{7}$

                         =  0.2668