13) Seven percent of Americans of type o- blood. You randomly select 30 Americans and ask them if their blood type is o- .
a. Find the probability that exactly 10 people say they have o- blood.
b. Find the probability that at least 10 people say they have o- blood.
c. Find the probability that fewer than 10 people say they have o- blood.
We can solve these questions using the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
where n is the number of trials, p is the probability of success, k is the number of successes, and nCk is the combination formula (n choose k).
a. Find the probability that exactly 10 people say they have o- blood.
Here, n = 30, p = 0.07, and k = 10.
P(X = 10) = 30C10 * (0.07)^10 * (1-0.07)^(30-10)
P(X = 10) = 30045015 * (0.07)^10 * (0.93)^20
P(X = 10) ≈ 0.0011
b. Find the probability that at least 10 people say they have o- blood.
This is the same as finding the probability that there are 10, 11, 12, ..., 30 people with o- blood.
P(X ≥ 10) = P(X = 10) + P(X = 11) + ... + P(X = 30)
To find this probability more easily, we can find the probability that there are fewer than 10 people with o- blood, and then subtract that from 1.
P(X ≥ 10) = 1 - P(X < 10)
P(X < 10) = P(X = 0) + P(X = 1) + ... + P(X = 9)
We can use the same binomial probability formula for each of these terms:
P(X < 10) ≈ 0.9995
Now, subtract this from 1 to get the probability of at least 10 people having o- blood:
P(X ≥ 10) ≈ 1 - 0.9995 ≈ 0.0005
c. Find the probability that fewer than 10 people say they have o- blood.
We already found this probability in part b:
P(X < 10) ≈ 0.9995