Question

The brand name of a certain chain of coffee shops has a 53% recognition rate in the town of Coffeeton. An executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in the town. He selects a random sample of 7 Coffeeton residents. Find the probability that exactly 4 of the 7 Coffeeton residents recognize the brand name.A) 0.00819B) 287C) 0.0789D) 0.254

Answer 1

Answer:

B) 0.287

Step-by-step explanation:

For each Coffeeton resident, there are only two possible outcomes. Either they recognize the brand name, or they do not. The probability of a resident recognizing the brand name is independent of other residents. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

53% recognition rate in the town of Coffeeton.

This means that $p = 0.53$

Find the probability that exactly 4 of the 7 Coffeeton residents recognize the brand name.

This is P(X = 4) when n = 7. So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 4) = C_{7,4}.(0.53)^{4}.(0.47)^{3} = 0.287$

So the anwer is:

B) 0.287

Answer 2

Answer:

Probability that exactly 4 of the 7 Coffee ton residents recognize the brand name is 0.287.

Step-by-step explanation:

We are given that the brand name of a certain chain of coffee shops has a 53% recognition rate in the town of Coffee ton.

Also, he selects a random sample of 7 Coffee ton residents.

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 = 7 Coffee ton residents

            r = number of success = exactly 4

           p = probability of success which in our question is % of recognition

                 rate in the town of Coffee ton, i.e; 53%

LET X = Number of Coffee ton residents recognizing the brand name

So, it means X ~ $Binom(n=7, p=0.53)$

Now, Probability that exactly 4 of the 7 Coffee ton residents recognize the brand name is given by = P(X = 4)

        P(X = 4) =  $\binom{7}{4}\times 0.53^{4} \times (1-0.53)^{7-4}$

                      =  $35 \times 0.53^{4} \times 0.47^{3}$

                      = 0.287

                

Therefore, Probability that exactly 4 of the 7 Coffee ton residents recognize the brand name is 0.287.