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 questions, we can use the concept of binomial probability. In a binomial experiment:

- There are a fixed number of trials (in this case, 4 trials since there are 4 people).
- Each trial has exactly two possible outcomes, often referred to as success and failure.
- The probability of success is the same for each trial, denoted as p.

In this scenario, each person has a 50% chance of correctly identifying the gourmet coffee (since they are guessing). Therefore, p = 0.5.

Now let's calculate the probabilities for the given questions:

a. Probability that exactly 2 of the 4 correctly identify the gourmet coffee:

To calculate this probability, we need to determine the number of ways we can choose 2 people out of 4 who guess correctly. This can be calculated using the binomial coefficient, also known as the choose function (denoted as "C(n, r)").

The formula for the probability of exactly r successes in n trials is:
P(X = r) = C(n, r) * p^r * (1 - p)^(n - r)

In this case, r = 2, n = 4, and p = 0.5. Let's substitute these values into the formula:

P(X = 2) = C(4, 2) * (0.5)^2 * (1 - 0.5)^(4 - 2)

Using the binomial coefficient formula, we have:
C(4, 2) = 4! / (2!(4-2)!) = 6

Substituting the values into the formula, we get:
P(X = 2) = 6 * (0.5)^2 * (0.5)^2 = 6 * 0.25 * 0.25 = 0.375

Therefore, the probability that exactly 2 of the 4 correctly identify the gourmet coffee is 0.375.

b. Probability that at least 2 of the 4 correctly identify the gourmet coffee:

To calculate this probability, we need to consider all the possibilities where 2, 3, or 4 out of the 4 people guess correctly.

We already know the probability of exactly 2 people guessing correctly (calculated in part a). Now, let's calculate the probability of exactly 3 people guessing correctly and exactly 4 people guessing correctly.

P(X = 3) = C(4, 3) * (0.5)^3 * (1 - 0.5)^(4 - 3)
C(4, 3) = 4! / (3!(4-3)!) = 4

P(X = 3) = 4 * (0.5)^3 * (0.5)^1 = 4 * 0.125 * 0.5 = 0.25

P(X = 4) = C(4, 4) * (0.5)^4 * (1 - 0.5)^(4 - 4)
C(4, 4) = 4! / (4!(4-4)!) = 1

P(X = 4) = 1 * (0.5)^4 * (0.5)^0 = 1 * 0.0625 * 1 = 0.0625

To calculate the probability of "at least 2 people," we add the probabilities of these three events together:
Probability at least 2 = P(X = 2) + P(X = 3) + P(X = 4)
Probability at least 2 = 0.375 + 0.25 + 0.0625
Probability at least 2 = 0.6875

Therefore, the probability that at least 2 out of the 4 correctly identify the gourmet coffee is 0.6875.

c. Probability that at most 1 out of the 4 correctly identify the gourmet coffee:

To calculate this probability, we need to consider the probabilities of exactly 0 people guessing correctly and exactly 1 person guessing correctly.

P(X = 0) = C(4, 0) * (0.5)^0 * (1 - 0.5)^(4 - 0)
C(4, 0) = 4! / (0!(4-0)!) = 1

P(X = 0) = 1 * (0.5)^0 * (0.5)^4 = 1 * 1 * 0.0625 = 0.0625

P(X = 1) = C(4, 1) * (0.5)^1 * (1 - 0.5)^(4 - 1)
C(4, 1) = 4! / (1!(4-1)!) = 4

P(X = 1) = 4 * (0.5)^1 * (0.5)^3 = 4 * 0.5 * 0.125 = 0.25

To calculate the probability of "at most 1 person," we add the probabilities of these two events together:
Probability at most 1 = P(X = 0) + P(X = 1)
Probability at most 1 = 0.0625 + 0.25
Probability at most 1 = 0.3125

Therefore, the probability that at most 1 out of the 4 correctly identify the gourmet coffee is 0.3125.