Final answer:
To find the probability of finding a carton with at least 2 broken eggs in the first three cartons inspected, we can use the concept of complementary probability. The probability is approximately 0.0408.
Step-by-step explanation:
To find the probability that the quality control inspector finds a carton with at least 2 broken eggs in the first three cartons inspected, we can use the concept of complementary probability. First, let's find the probability that none of the three cartons have at least 2 broken eggs. To do this, we need to find the probability that each carton has 0 or 1 broken egg:
- In the first carton, the probability of 0 or 1 broken egg is (10/144) + (12/144) = 22/144 = 11/72.
- In the second carton, the probability of 0 or 1 broken egg given that the first carton had 0 or 1 broken egg is (10/143) + (12/143) = 22/143.
- In the third carton, the probability of 0 or 1 broken egg given that the first two cartons had 0 or 1 broken eggs is (10/142) + (12/142) = 22/142 = 11/71.
Now, we can find the probability that at least 2 broken eggs are found in the first three cartons by subtracting the probability of none of the cartons having 2 broken eggs from 1:
Probability at least 2 broken eggs in first three cartons = 1 - (11/72) * (22/143) * (11/71) = 101/2478 ≈ 0.0408 (rounded to 4 decimal places).