Final answer:
To find the probability of more than 37, but fewer than 40 people buying the food item, we can use the binomial distribution. Calculate the probabilities for each number of successes using the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k). Sum the probabilities for 38 and 39 people to get the total probability.
Step-by-step explanation:
To find the probability that more than 37, but fewer than 40 people in a sample of 50 would buy a food item after tasting a free sample, we need to use the binomial distribution. The binomial distribution can be used when we have a fixed number of trials, each with two possible outcomes, and the probability of success remains the same for each trial.In this case, the probability of someone buying the food item after tasting the sample is not given, so we'll assume it to be p. The probability of someone not buying the food item would then be 1 - p.The formula for the probability of getting exactly k successes in n trials is P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) represents the number of ways to choose k items from a set of n.Since we want to find the probability of more than 37, but fewer than 40 people buying the food item, we need to calculate P(X = 38) + P(X = 39).Let's use an example where p = 0.5 to illustrate the calculation:
P(X = 38) = C(50, 38) * 0.5^38 * (1-0.5)^(50-38) = (50 choose 38) * 0.5^38 * 0.5^12
P(X = 39) = C(50, 39) * 0.5^39 * (1-0.5)^(50-39) = (50 choose 39) * 0.5^39 * 0.5^11
Finally, we can sum the two probabilities to get the total probability that more than 37, but fewer than 40 people would buy the food item.