Final answer:
To find the probability that at least 10 eggs in a randomly selected carton are unbroken, we need to use the hypergeometric distribution. The probability can be found by summing the probabilities of having 0, 1, 2, or 3 cracked eggs and taking the complement.
Step-by-step explanation:
To find the probability that at least 10 eggs in a randomly selected carton are unbroken, we need to use the concept of hypergeometric distribution. In this case, we have a total of 144 eggs in a gross, and we know that 12 eggs are cracked. The inspector randomly chooses 15 eggs for inspection.
The probability that at most three eggs are cracked can be found by summing the probabilities of having 0, 1, 2, or 3 cracked eggs. We can use the following formula:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
P(X = 0) can be calculated using the hypergeometric probability formula as:
P(X = 0) = (C(132, 15) * C(12, 0)) / C(144, 15)
Similarly, you can calculate P(X = 1), P(X = 2), and P(X = 3) using the same formula. Once you have the individual probabilities, you can add them up to find the probability that at most three eggs are cracked. The answer will be the complement of this probability, as we are looking for the probability that at least 10 eggs are unbroken.