it is estimated that 17% of americans have blue eyes. A random sample of 9 americans is selected. Find the probability that the sample includes exactly 2 people with blue eyes.
9C2(.17)^2(.83)^7
To find the probability that the sample includes exactly 2 people with blue eyes, you can use the binomial probability formula:
P(X = k) = (n C k) * (p^k) * ((1-p)^(n-k))
Where:
- n is the total number of trials (sample size)
- k is the number of successes (number of people with blue eyes)
- p is the probability of success (proportion of Americans with blue eyes)
In this case, n = 9 (sample size), k = 2 (number of people with blue eyes), and p = 0.17 (proportion of Americans with blue eyes).
Using the binomial probability formula, the probability can be calculated as follows:
P(X = 2) = (9 C 2) * (0.17^2) * ((1-0.17)^(9-2))
First, calculate (9 C 2), which represents the number of ways to choose 2 people out of a sample of 9:
(9 C 2) = (9!)/[(2!)(9-2)!] = (9!)/(2! x 7!) = (9 x 8)/(2 x 1) = 36
Next, substitute the values into the formula:
P(X = 2) = 36 * (0.17^2) * (0.83^7)
Now, calculate (0.17^2) and (0.83^7):
0.17^2 = 0.0289
0.83^7 = 0.2925
Substitute these values back into the formula:
P(X = 2) = 36 * 0.0289 * 0.2925 = 0.0308 (rounded to four decimal places)
Therefore, the probability that the sample includes exactly 2 people with blue eyes is approximately 0.0308, or 3.08%.