Ecstasy Products sells a certain product to pharmacies in...

Ecstasy Products sells a certain product to pharmacies in boxes of 100 with a guarantee that at most 10 items in a box are defective. Debilitating Drugs has a buyer who accepts a box of the product only if a random sample of 10 items chosen without replacement from the box contains no defective items. What is the probability that a box will be rejected although it contains exactly 10 defective items and thus meets the conditions of the guarantee?

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution is a probability distribution used when sampling without replacement from a finite population that contains two types of items. It calculates the likelihood of a given number of successes (such as selecting non-defective items) in a sample drawn from a population with a fixed number of successes and failures. In this context, it is used to determine the probability of drawing a sample with zero defective items from a box containing both defective and non-defective items.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways events can occur, often using combinations. In problems like this, combinations are used to calculate the number of ways to choose items from a population. For example, calculating the number of ways to select a sample of items that meets certain criteria (such as being all non-defective) relative to the total number of possible samples informs the probability of that event.

Complementary Probability

Complementary probability is a method of finding the probability of an event by subtracting the probability of the event's complement from one. In sampling scenarios, it is often easier to compute the probability of a favorable outcome (like drawing only non-defective items) and then subtract that from one to obtain the probability of the unfavorable outcome (such as having at least one defective item in the sample).

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