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 (b) one microchip is defective and two are not defective

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that one microchip is defective and two are not defective using the hypergeometric distribution formula. The formula is: $P_x=\frac{\left(k_x\right)\left(N_{-}\right)}{N_n}$, 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 = 1 (number of defective microchips in the sample).
3. Step 3: Break down the formula. The numerator consists of two combinations: $k_x$ represents the number of ways to choose x defective microchips from k defective ones, and $N_{-}$ represents the number of ways to choose the remaining (n - x) non-defective microchips from the non-defective ones. The denominator is $N_n$, which represents the total number of ways to choose n microchips from the population.
4. Step 4: Calculate each combination. Use the combination formula $\left(a_b\right)=\frac{a!}{b!(a-b)!}$. For $k_x$, calculate $(2!)$. For $N_{-}$, calculate $(13!)$. Finally, calculate $N_n$ for the total population.
5. Step 5: Plug the values into the formula. Substitute the calculated combinations into the hypergeometric formula and simplify. This will give the probability of selecting one defective microchip and two non-defective microchips in the sample.

Key Concepts

Hypergeometric Distribution

The hypergeometric distribution models the probability of obtaining a specific number of successes in a sample drawn without replacement from a finite population. Unlike the binomial distribution, where sampling is done 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).

Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. In the context of the hypergeometric distribution, combinations are used to calculate the number of ways to choose successes and failures from the population. The notation 'nCr' represents the number of combinations of n items taken r at a time, calculated using the formula n! / (r!(n-r)!), where '!' denotes factorial.

Probability Calculation

Probability calculation in the hypergeometric distribution involves determining the likelihood of a specific outcome based on the number of successes and failures in the sample. The formula provided combines the combinations of successes and failures in the sample and the population, divided by the total combinations possible in the population. This allows for the computation of the probability of drawing a certain number of defective and non-defective items from the shipment of microchips.