Question

A lot consisting of 100 fuses, is inspected by the following Procedure. Five of these fuses are chosen at random and tested; if all 5 "blow" at the correct amperage, the lot is accepted. Find the probability distribution of the number of defectives in a sample of 5 assuming there are 20 in the lot.

Short answer

The probability distribution of the number of defectives in a sample of 5 fuses, assuming there are 20 defective fuses in the total lot of 100 fuses, is given by: P(X = 0) = \(\frac{\binom{20}{0}\binom{80}{5}}{\binom{100}{5}}\), P(X = 1) = \(\frac{\binom{20}{1}\binom{80}{4}}{\binom{100}{5}}\), P(X = 2) = \(\frac{\binom{20}{2}\binom{80}{3}}{\binom{100}{5}}\), P(X = 3) = \(\frac{\binom{20}{3}\binom{80}{2}}{\binom{100}{5}}\), P(X = 4) = \(\frac{\binom{20}{4}\binom{80}{1}}{\binom{100}{5}}\), P(X = 5) = \(\frac{\binom{20}{5}\binom{80}{0}}{\binom{100}{5}}\).

Step-by-step solution

Identify the parameters

The given parameters in our problem are as follows: - Total number of fuses in the lot (N): 100 - Number of defective fuses (K): 20 - Sample size (n): 5

Define the probability function

In this step, we define the probability function using the hypergeometric distribution. For a given number of defective fuses, k, the probability is given by: P(X = k) = \(\frac{\binom{20}{k}\binom{80}{5-k}}{\binom{100}{5}}\)

Calculate the probabilities

Now, we will calculate the probability distribution for each possible number of defective fuses, k, in our sample of 5 fuses: 1. P(X = 0): Prob. of no defective fuses in the sample P(X = 0) = \(\frac{\binom{20}{0}\binom{80}{5-0}}{\binom{100}{5}}\) 2. P(X = 1): Prob. of exactly 1 defective fuse in the sample P(X = 1) = \(\frac{\binom{20}{1}\binom{80}{5-1}}{\binom{100}{5}}\) 3. P(X = 2): Prob. of exactly 2 defective fuses in the sample P(X = 2) = \(\frac{\binom{20}{2}\binom{80}{5-2}}{\binom{100}{5}}\) 4. P(X = 3): Prob. of exactly 3 defective fuses in the sample P(X = 3) = \(\frac{\binom{20}{3}\binom{80}{5-3}}{\binom{100}{5}}\) 5. P(X = 4): Prob. of exactly 4 defective fuses in the sample P(X = 4) = \(\frac{\binom{20}{4}\binom{80}{5-4}}{\binom{100}{5}}\) 6. P(X = 5): Prob. of all 5 fuses being defective in the sample P(X = 5) = \(\frac{\binom{20}{5}\binom{80}{5-5}}{\binom{100}{5}}\)

Final probability distribution

The probability distribution of the number of defectives in a sample of 5 fuses, assuming there are 20 defective fuses in the total lot of 100 fuses, is given by: P(X = 0) = \(\frac{\binom{20}{0}\binom{80}{5}}{\binom{100}{5}}\), P(X = 1) = \(\frac{\binom{20}{1}\binom{80}{4}}{\binom{100}{5}}\), P(X = 2) = \(\frac{\binom{20}{2}\binom{80}{3}}{\binom{100}{5}}\), P(X = 3) = \(\frac{\binom{20}{3}\binom{80}{2}}{\binom{100}{5}}\), P(X = 4) = \(\frac{\binom{20}{4}\binom{80}{1}}{\binom{100}{5}}\), P(X = 5) = \(\frac{\binom{20}{5}\binom{80}{0}}{\binom{100}{5}}\).

Key concepts

Probability Distribution

In statistics, a probability distribution is a fundamental concept that describes how the probabilities are distributed across the possible outcomes of a random experiment. It is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.
For discrete random variables, like the number of defective fuses in our example, the probability distribution is specifically called a discrete probability distribution. It lists the probabilities of each possible outcome of a discrete random variable, and these probabilities must sum up to one. Understanding the distribution helps in visualizing the likelihood of various outcomes and is vital in fields such as finance, insurance, and engineering, where predictive models based on probability are essential.

Combinatorics

At the heart of solving problems involving the hypergeometric distribution lies combinatorics, the branch of mathematics dealing with counting, combinations, permutations, and the structure of possible arrangements. In the context of our fuse example, combinatorics helps to determine the number of ways defective fuses can be chosen from the lot.
Specifically, we use combinations — denoted as \( \binom{n}{k} \) — which calculate how many ways we can select k items from a set of n items without regard to the order. The solutions involve using combinations to calculate the probabilities of obtaining different numbers of defective fuses in the sample. This understanding of combinatorial mathematics not only helps in calculating probabilities but also in various areas like cryptography, algorithms, and even game theory.

Quality Control

In manufacturing and business processes, quality control is the methodology of ensuring products meet customer expectations. The hypergeometric distribution is instrumental in quality control because it helps in decision-making based on a sample of products from a larger lot. Take our fuse example, by examining just five fuses out of a hundred, the quality control process attempts to infer the quality of the entire lot.
By determining the probability distribution of defective fuses in the sample, quality control personnel can estimate the reliability of the lot and thus decide whether a lot should be accepted or rejected. This practice is common in industries where testing all items is impractical or destructive, and it underscores the importance of statistical methods in ensuring quality, safety, and customer satisfaction.

Discrete Random Variables

A discrete random variable is a type of random variable that can take on a countable number of distinct outcomes. For instance, the number of defective fuses within a sample falls under this category, as it can only be integers from 0 to 5.
The significance of understanding discrete random variables stems from their predictability and analysis in probability distributions. Every outcome has a specific probability associated with it, as demonstrated in our fuse sample problem. Managers and decision-makers use discrete random variables to calculate risks and make informed decisions based on the calculated probabilities. They are integral in sectors ranging from telecommunications to healthcare, where discrete events like clicks or patient arrivals are analyzed.