Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and failures, the probability of selecting a sample of size that has successes and failures is given by



In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (c) two microchips are defective and one is not defective.

Verified step by step guidance

1. Step 1: Understand the hypergeometric distribution formula. The formula is given as: $P(x)\; =\frac{\left(k_C^x\right)\left(N_{-}^k{C}_n^{-}\right)}{N_C^n}$, where N is the population size, k is the number of successes in the population, n is the sample size, and x is the number of successes in the sample.
2. Step 2: Identify the values from the problem. Here, N = 15 (total microchips), k = 2 (defective microchips), n = 3 (sample size), and x = 2 (number of defective microchips in the sample).
3. Step 3: Compute the numerator of the formula. This involves calculating two combinations: $k_C^x$ and $N_{-}^k{C}_n^{-}$. Specifically, calculate $2_C^2$ and ${13}_C^1$.
4. Step 4: Compute the denominator of the formula. This involves calculating the combination ${15}_C^3$, which represents the total number of ways to choose 3 microchips from the shipment.
5. Step 5: Substitute the computed values into the formula and simplify. Divide the product of the numerator combinations by the denominator combination to find the probability that two microchips are defective and one is not defective.

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. Unlike the binomial distribution, which assumes independent trials with replacement, the hypergeometric distribution accounts for the changing probabilities as items are drawn. It is defined by the population size (N), the number of successes in the population (k), the sample size (n), and the number of successes in the sample (x).

Combinatorial Notation

Combinatorial notation, often represented as 'n choose k' or C(n, k), is used to calculate the number of ways to choose k successes from n trials. This is crucial in the hypergeometric distribution formula, where it helps determine the number of ways to select successes and failures from the population. Understanding how to compute combinations is essential for solving problems involving the hypergeometric distribution.

Probability Calculation

Probability calculation in the context of the hypergeometric distribution involves determining the likelihood of a specific outcome based on the defined parameters. The formula provided combines the combinations of successes and failures to yield the probability of obtaining a certain number of defective items in a sample. This requires substituting the values of N, k, n, and x into the formula to find the desired probability.