Math  /  Data & Statistics

QuestionA study showed that 8%\mathbf{8 \%} of American teenagers have tattoos. 28\mathbf{2 8} teenagers are randomly selected. What is the probability that exactly 2\mathbf{2} will have a tattoo? (Round the answer up to 4 decimal places) \square Done

Studdy Solution

STEP 1

1. The probability of a teenager having a tattoo is p=0.08 p = 0.08 .
2. The number of teenagers selected is n=28 n = 28 .
3. We are looking for the probability that exactly 2 teenagers have tattoos.
4. This is a binomial probability problem.

STEP 2

1. Identify the binomial probability formula.
2. Substitute the given values into the formula.
3. Calculate the probability.
4. Round the result to four decimal places.

STEP 3

The binomial probability formula is:
P(X=k)=(nk)pk(1p)nk P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
where n n is the number of trials, k k is the number of successful trials, p p is the probability of success, and (nk)\binom{n}{k} is the binomial coefficient.

STEP 4

Substitute the given values into the formula:
n=28,k=2,p=0.08 n = 28, \, k = 2, \, p = 0.08
P(X=2)=(282)(0.08)2(10.08)282 P(X = 2) = \binom{28}{2} (0.08)^2 (1-0.08)^{28-2}

STEP 5

Calculate the binomial coefficient:
(282)=28×272×1=378 \binom{28}{2} = \frac{28 \times 27}{2 \times 1} = 378
Calculate the probability:
P(X=2)=378×(0.08)2×(0.92)26 P(X = 2) = 378 \times (0.08)^2 \times (0.92)^{26}
P(X=2)=378×0.0064×0.1304 P(X = 2) = 378 \times 0.0064 \times 0.1304
P(X=2)378×0.00083584 P(X = 2) \approx 378 \times 0.00083584
P(X=2)0.3169 P(X = 2) \approx 0.3169

STEP 6

Round the result to four decimal places:
P(X=2)0.3169 P(X = 2) \approx 0.3169
The probability that exactly 2 teenagers will have a tattoo is:
0.3169 \boxed{0.3169}

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