Each of 4 people is presented with gourmet coffee and ordinary coffee and asked to identify the gourmet coffee. None of the 4 can really tell the difference and are merely guessing. a. What is the probability that exactly 2 of the 4 correctly identify the gourmet coffee? b. What is the probability that at least 2 of the 4 correctly identify the gourmet coffee? c. What is the probability that at most 1 out of the 4 correctly identify the gourmet coffee?
To solve these probability problems, we can use the concept of binomial probability.
Binomial probability calculates the probability of a certain number of successes in a fixed number of independent Bernoulli trials.
In this case, each person's guess is an independent Bernoulli trial with a probability of success (correctly identifying the gourmet coffee) of 1/2 (since they are merely guessing).
a. To calculate the probability that exactly 2 people correctly identify the gourmet coffee, we can use the binomial probability formula:
P(X=k) = nCk * p^k * q^(n-k)
where:
P(X=k) is the probability of getting exactly k successes,
n is the total number of trials,
k is the number of successes,
p is the probability of success in a single trial, and
q is the probability of failure in a single trial (1-p).
In this case, n = 4, k = 2, p = 1/2, and q = 1/2.
Using the formula, we can calculate:
P(X=2) = 4C2 * (1/2)^2 * (1/2)^(4-2)
P(X=2) = 6 * 1/4 * 1/4
P(X=2) = 6/16
P(X=2) = 3/8
Therefore, the probability that exactly 2 of the 4 people correctly identify the gourmet coffee is 3/8.
b. To calculate the probability that at least 2 people correctly identify the gourmet coffee, we need to find the sum of the probabilities of getting 2, 3, and 4 successes.
P(at least 2) = P(X=2) + P(X=3) + P(X=4)
We already know P(X=2) from part a. Now, let's calculate P(X=3):
P(X=3) = 4C3 * (1/2)^3 * (1/2)^(4-3)
P(X=3) = 4 * 1/8 * 1/2
P(X=3) = 1/4
Finally, let's calculate P(X=4):
P(X=4) = 4C4 * (1/2)^4 * (1/2)^(4-4)
P(X=4) = 1 * 1/16 * 1
P(X=4) = 1/16
Adding all these probabilities together:
P(at least 2) = 3/8 + 1/4 + 1/16
P(at least 2) = 12/32 + 8/32 + 2/32
P(at least 2) = 22/32
P(at least 2) = 11/16
Therefore, the probability that at least 2 of the 4 people correctly identify the gourmet coffee is 11/16.
c. To calculate the probability that at most 1 out of the 4 people correctly identify the gourmet coffee, we can subtract the probability of getting exactly 2, 3, and 4 successes from 1 (the total probability).
P(at most 1) = 1 - (P(X=2) + P(X=3) + P(X=4))
We already know P(X=2) and P(X=3) from parts a and b. Now, let's calculate P(X=4):
P(X=4) = 1/16 (same as in part b)
Substituting the values:
P(at most 1) = 1 - (3/8 + 1/4 + 1/16)
P(at most 1) = 1 - (6/16 + 4/16 + 1/16)
P(at most 1) = 1 - 11/16
P(at most 1) = 5/16
Therefore, the probability that at most 1 out of the 4 people correctly identify the gourmet coffee is 5/16.