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. Find the probability that the sample includes at most 2 people with blue eyes. Find the mean number of blu-eyed people in a sample of 9 americans.
Mean = np = 9 * .17 = 1.53
Use binomial probability formula (or a binomial probability table).
Formula:
P(x) = (nCx)(p^x)[q^(n-x)]
x = 0,1,2
n = 9
p = .17
q = 1 - p = 1 - .17 = .83
For first part: Find P(2) for probability.
For second part: Find P(0) and P(1). Add P(0), P(1), and P(2) for probability.
I'll let you take it from here.
To find the probabilities and mean number of blue-eyed people in a sample, we can use the concept of binomial probability. In this case, we have a binomial distribution because we are interested in the number of successes (people with blue eyes) in a fixed number of trials (sample of 9 Americans).
To find the probability that the sample includes exactly 2 people with blue eyes, we use the formula for binomial probability:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) is the probability of getting exactly k successes (people with blue eyes)
- n is the number of trials (sample size)
- k is the number of successes we want
- p is the probability of success in each trial (probability of an American having blue eyes)
- (n choose k) is the binomial coefficient, representing the number of ways to choose k successes from n trials
Plugging in the values, we have:
- n = 9 (sample size)
- k = 2 (number of people with blue eyes)
- p = 0.17 (probability of an American having blue eyes)
P(X = 2) = (9 choose 2) * 0.17^2 * (1 - 0.17)^(9 - 2)
To calculate this, we use the binomial coefficient formula:
(9 choose 2) = 9! / (2! * (9 - 2)!) = 36
After substitution, we get:
P(X = 2) = 36 * 0.17^2 * 0.83^7
Simplifying and calculating, we find:
P(X = 2) ≈ 0.294
Therefore, the probability that the sample includes exactly 2 people with blue eyes is approximately 0.294.
Moving on to the probability of the sample including at most 2 people with blue eyes, we need to find the probabilities for 0, 1, and 2 people with blue eyes, and then sum them up.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
Using the same formula as before, we calculate each probability and add them up.
P(X = 0) = (9 choose 0) * 0.17^0 * 0.83^9
= 1 * 1 * 0.83^9
P(X = 1) = (9 choose 1) * 0.17^1 * 0.83^8
= 9 * 0.17 * 0.83^8
We have already calculated P(X = 2) as approximately 0.294.
Summing these probabilities:
P(X ≤ 2) = 0.83^9 + 9 * 0.17 * 0.83^8 + 0.294
≈ 0.903
Therefore, the probability that the sample includes at most 2 people with blue eyes is approximately 0.903.
To find the mean number of blue-eyed people in a sample of 9 Americans, we can use the formula for the mean of a binomial distribution:
Mean (μ) = n * p
Plugging in the values, we get:
Mean (μ) = 9 * 0.17
= 1.53
Therefore, the mean number of blue-eyed people in a sample of 9 Americans is 1.53.