Question

A bottle of a certain product is to be filled to the correct height. Historically, 3% of the filled bottles are underfilled. A quality control inspector randomly selects 10 bottles from the production line. How likely is it that there are more than 2 underfilled bottles in a sample set?

Answer 1

To find the probability of having more than two underfilled bottles in a sample of 10, we can use the binomial probability formula. Since manually summing the probabilities for 3 to 10 underfilled bottles is complex, we calculate the complementary probability and subtract it from 1.

Step-by-step explanation:

The likelihood of there being more than two underfilled bottles in a sample set can be calculated using the binomial probability formula. Considering that there's historically a 3% chance that a bottle is underfilled, we want to find the probability of having more than 2 underfilled bottles in a sample set of 10. To compute this, we need to find the sum of the probabilities of there being 3, 4, 5, ..., up to 10 underfilled bottles, which can be a bit cumbersome manually. Hence, it is often more convenient to calculate the complementary probability (0, 1, or 2 underfilled bottles) and subtract it from 1 to get the desired probability. The binomial probability formula used for each case is P(X=k) = C(n,k) * p^k * (1-p)^(n-k) where C(n,k) denotes the combination of n items taken k at a time, p is the probability of a single event, and n is the total number of events.