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 need to first understand the concept of combinations. The formula for finding the number of combinations is given by nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being selected.
For the given problem, there are 4 people, each of whom can choose either the gourmet coffee or the ordinary coffee. Since they are guessing, we can assume there are only two options for each person.
a) The probability of exactly 2 people correctly identifying the gourmet coffee can be calculated as follows:
- First, we need to find the number of ways to select 2 people out of 4. This can be calculated as 4C2 = 4! / (2!(4-2)!) = 6.
- The probability that a person correctly guesses the gourmet coffee is 1/2, since they are guessing.
- The probability that a person incorrectly guesses the gourmet coffee is also 1/2.
- Therefore, the probability that exactly 2 people correctly identify the gourmet coffee is (1/2)^2 * (1/2)^2 * 6 = 1/16.
b) The probability that at least 2 people correctly identify the gourmet coffee can be calculated using the concept of complement:
- The probability that at least 2 people correctly identify the gourmet coffee is equal to 1 - the probability that less than 2 people correctly identify the gourmet coffee.
- The probability that exactly 0 people correctly identify the gourmet coffee is (1/2)^4 = 1/16 (since each person is guessing independently).
- The probability that exactly 1 person correctly identifies the gourmet coffee can be calculated as follows:
- The number of ways to choose 1 person out of 4 is 4C1 = 4.
- The probability that this person correctly guesses the gourmet coffee is 1/2.
- The probability that the rest of the people incorrectly guess the gourmet coffee is (1/2)^3 = 1/8.
- Therefore, the probability that exactly 1 person correctly identifies the gourmet coffee is 4 * (1/2) * (1/8) = 1/4.
- Thus, the probability that less than 2 people correctly identify the gourmet coffee is 1/16 + 1/4 = 5/16.
- Finally, the probability that at least 2 people correctly identify the gourmet coffee is 1 - 5/16 = 11/16.
c) The probability that at most 1 person correctly identifies the gourmet coffee can be calculated by subtracting the probability of at least 2 people correctly identifying the gourmet coffee from 1.
- The probability that at most 1 person correctly identifies the gourmet coffee is equal to 1 - the probability that at least 2 people correctly identify the gourmet coffee.
- We already calculated that the probability of at least 2 people correctly identifying the gourmet coffee is 11/16.
- Therefore, the probability that at most 1 person correctly identifies the gourmet coffee is 1 - 11/16 = 5/16.
So, to summarize:
a) The probability that exactly 2 of the 4 correctly identify the gourmet coffee is 1/16.
b) The probability that at least 2 of the 4 correctly identify the gourmet coffee is 11/16.
c) The probability that at most 1 out of the 4 correctly identify the gourmet coffee is 5/16.