Mathematics: According to a recent New York Times poll, 25% of the public have responded to a telephone call-in poll. Based upon this information, what is the probability that exactly two people out of a randomly chosen group of five people have responded to a telephone call-in poll? Round your answer to the nearest thousandth (3 decimal places).

According to a recent New York Times poll, 25% of

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26.01.2022

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.

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