Question

The makers of a diet cola claim that its taste is indistinguishable from the full-calorie

version of the same cola. To investigate, a student names Amy prepared small
samples of each type of soda in identical cups. Then she had 15 volunteers taste
each cola in a random order to try to identify which was the diet cola and which was
the regular cola. If we assume that the volunteers couldn't tell the difference, then
each one was guessing with a ½ chance of being correct. Let Y= the number
volunteers who correctly identify the colas. Use the binomial probability formula or
your calculator to find the following.
P(Y <3)
Round your answer to the nearest thousandth.

Answer 1

The probability of each volunteer guessing correctly is 1/2, since there are only two options and they are guessing randomly. We can use the binomial probability formula to calculate the probability of Y volunteers correctly identifying the colas:

P(Y=k) = (n choose k) * p^k * (1-p)^(n-k)

where:
- n is the number of trials (15 in this case)
- k is the number of successes (in this case, the number of volunteers who correctly identify the colas)
- p is the probability of success on each trial (1/2)

To find P(Y ≤ 3), we need to calculate the probability of Y=0, Y=1, Y=2, or Y=3. We can do this by plugging in k=0, 1, 2, and 3 into the binomial probability formula and summing the results:

P(Y ≤ 3) = P(Y=0) + P(Y=1) + P(Y=2) + P(Y=3)
= (15 choose 0) * (1/2)^0 * (1/2)^15 + (15 choose 1) * (1/2)^1 * (1/2)^14 + (15 choose 2) * (1/2)^2 * (1/2)^13 + (15 choose 3) * (1/2)^3 * (1/2)^12
≈ 0.003

Rounding to the nearest thousandth, we get:

P(Y ≤ 3) ≈ 0.003