Question

A recent survey of U.S. adults found that 75% would pay more for an environmentally friendly product. You randomly select 12 adults and ask them if they would pay more for environmentally friendly products. Find the probability that no more than ten adults say that they would pay more for environmentally friendly products. Round your answer to three decimal places.

Answer 1

Probability that no more than ten adults say that they would pay more for environmentally friendly products is 0.842.

Step-by-step explanation:

We are given that a recent survey of U.S. adults found that 75% would pay more for an environmentally friendly product.

We randomly select 12 adults and ask them if they would pay more for environmentally friendly products.

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 = 12 adults

            r = number of success = not more than 10

           p = probability of success which in our question is in a survey % of

                 U.S. adults who would pay more for an environmentally friendly

                  product . i.e; 75%

LET X = Number of adults who would pay more for environmentally friendly products

So, it means X ~ $Binom(n=12, p=0.75)$

Now, Probability that no more than ten adults say that they would pay more for environmentally friendly products is given by = P(X $\leq$ 10)

 P(X $\leq$ 10) = 1 - P(X > 10)

                 = 1 - [ P(X = 11) + P(X = 12) ]

                = 1 - [ $\binom{12}{11}\times 0.75^{11} \times (1-0.75)^{12-11}+\binom{12}{12} \times 0.75^{12} \times (1-0.75)^{12-12}$ ]

                = 1 - [ $12 \times 0.75^{11} \times 0.25^{1} + 1 \times 0.75^{12} \times 1$ ]

                = 1 - 0.1584 = 0.842

Therefore, probability that no more than ten adults say that they would pay more for environmentally friendly products is 0.842.