Question

In a recent survey, 60% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 5 of them favor the building of the health center.

Answer 1

Answer:

Probability that exactly 5 of them favor the building of the health center is 0.0408.

Step-by-step explanation:

We are given that in a recent survey, 60% of the community favored building a health center in their neighborhood.

Also, 14 citizens are chosen.

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 = 14 citizens

         r = number of success = exactly 5

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

              favored building a health center in their neighborhood, i.e; 60%

LET X = Number of citizens who favored building of the health center.

So, it means X ~ $Binom(n=14, p=0.60)$

Now, Probability that exactly 5 of them favor the building of the health center is given by = P(X = 5)

        P(X = 5) = $\binom{14}{5} \times 0.60^{5} \times (1-0.60)^{14-5}$

                      = $2002 \times 0.60^{5} \times 0.40^{9}$

                      = 0.0408

Therefore, Probability that exactly 5 of them favor the building of the health center is 0.0408.