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