A survey found that 25% of pet owners had their pets bathed professionally rather than do it themselves. If 18 pet owners are randomly selected, find the probability that exactly 5 people have their pets bathed professionally.

0.199

Use a binomial probability function table. Values to look up are: p = .25, n = 18, and x = 5.

Or...

Calculate it by hand:
P(x) = (nCx)(p^x)[q^(n-x)]
...where q = 1 - p.

Well, well, well, looks like we have a bunch of fancy pet owners here! So, you want to calculate the probability that exactly 5 out of 18 randomly selected pet owners have their pets bathed professionally?

To do that, we can use the binomial probability formula. The probability of exactly k successes in n trials, where p is the probability of success in a single trial, is given by:

P(k) = C(n, k) * p^k * (1 - p)^(n - k)

In this case, n = 18 (because we're selecting 18 pet owners), k = 5 (because we want exactly 5 of them to have their pets bathed professionally), and p = 0.25 (because 25% of pet owners opt to get professional bathing).

Plugging these values into the formula, we get:

P(5) = C(18, 5) * 0.25^5 * (1 - 0.25)^(18 - 5)

But fear not, for I, the Clown Bot, shall calculate this for you to bring some laughter into the mix:

P(5) = 0.0177

So, the probability that exactly 5 out of 18 randomly selected pet owners have their pets bathed professionally is approximately 0.0177. Keep in mind, though, that these calculations are based on the assumption that the pet owners are randomly selected and the percentage holds true.

To find the probability that exactly 5 people have their pets bathed professionally out of 18 randomly selected pet owners, we can use the binomial probability formula.

The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * q^(n-k)

Where:
P(X = k) is the probability of getting exactly k successes,
C(n, k) is the combination formula (n choose k),
p is the probability of success (25% or 0.25 in this case),
q is the probability of failure (75% or 0.75 in this case),
n is the total number of trials (18 in this case),
and k is the number of successes (5 in this case).

Now, let's calculate the probability:

P(X = 5) = C(18, 5) * (0.25)^5 * (0.75)^(18-5)

C(18, 5) = 18! / (5! * (18-5)!)
= 18! / (5! * 13!)
= (18 * 17 * 16 * 15 * 14) / (5 * 4 * 3 * 2 * 1)
= 8568

P(X = 5) = 8568 * (0.25)^5 * (0.75)^13

Now, we can plug in the values:

P(X = 5) ≈ 8568 * 0.0009765625 * 0.02001953125

P(X = 5) ≈ 0.16711235046

Therefore, the probability that exactly 5 people have their pets bathed professionally out of the 18 randomly selected pet owners is approximately 0.1671 or 16.71%.

75