I'm not sure where to start...

Mail order marketing companies have a response rate of 15% to their advertising flyers. If 25 people receive the flyer, what is the probability that 5 people respond to the flyer?

Looks like a binomial distribution question.

I read your question to say that exactly 5 are to respond.

prob(response) = .15
prob(no response) = .85

prob(5 of 15 respond)
= C(15,5) (.15)^5 (.85)^10
= appr .0449

Where did you get the 10?

Oh, I see. There are 25 to receive the flyer total, you misread it as 15.

=C(25,5)(.15)^5(.85)^20
= appr 0.156

To determine the probability of 5 people responding to the flyer, we need to use the binomial probability formula. The formula is:

P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))

Where:
P(X = k) is the probability of X (in this case, the number of people responding) being equal to k (in this case, 5).
n is the total number of trials (25 people receiving the flyer).
k is the number of successful outcomes (5 people responding).
p is the probability of a single success (response rate, 15% or 0.15).
(1-p) is the probability of a single failure (not responding, 85% or 0.85).
nCk is the combination, which is calculated as n! / (k! * (n-k)!)

Now let's calculate the probability:

Step 1: Calculate nCk:
nCk = 25! / (5! * (25-5)!)
= 25! / (5! * 20!)
= (25 * 24 * 23 * 22 * 21) / (5 * 4 * 3 * 2 * 1)
= 53,130

Step 2: Calculate p^k:
p^k = 0.15^5
= 0.0000759375

Step 3: Calculate (1-p)^(n-k):
(1-p)^(n-k) = (1-0.15)^(25-5)
= 0.85^20
≈ 0.0282475249

Step 4: Multiply the results from steps 1, 2, and 3:
P(X = 5) = 53,130 * 0.0000759375 * 0.0282475249
≈ 0.1207

Therefore, the probability that exactly 5 people will respond to the flyer is approximately 0.1207 or 12.07%.