Question

According to a recent New York Times poll, 25% of the public have responded to a telephone call-in

Answer 1

Answer:

Probability that exactly two people out of a randomly  chosen group of five people have responded to a telephone call-in poll is 0.264.

Step-by-step explanation:

We are given that according to a recent New York Times poll, 25% of the public have responded to a telephone call-in  poll.

Also, five people have people have been randomly 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 trials (samples) taken = 5 people

           r = number of success = exactly two

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

                 public have responded to a telephone call-in  poll, i.e; p = 25%

Let X = Number of people who have responded to a telephone call-in  poll

So, X ~ Binom(n = 5, p = 0.25)

Now, probability that exactly two people out of group of five people have responded to a telephone call-in poll is given by = P(X = 2)

                P(X = 2)  =  $\binom{5}{2} \times 0.25^{2} \times (1-0.25)^{5-2}$

                               =  $10 \times 0.25^{2} \times 0.75^{3}$

                               =  0.264

Therefore, the probability that exactly two people out of a randomly  chosen group of five people have responded to a telephone call-in poll is 0.264.