A piece of electronic equipment contains six computer chips,...

A piece of electronic equipment contains six computer chips, two of which are defective. Three chips are selected at random, removed from the piece of equipment, and inspected. Let $x$ equal the number of defectives observed, where $x=0,1,$ or 2 . Find the probability distribution for $x$ Express the results graphically as a probability histogram.

Key Concepts

Graphical Representation Using a Probability Histogram

A probability histogram is a graphical tool used to display the distribution of a discrete random variable. It represents each outcome (or value of the variable) with a bar whose height corresponds to the probability of that outcome according to the PMF. This visual representation helps in understanding how probabilities are distributed across the possible outcomes.

Hypergeometric Distribution

This distribution models the probability of drawing a given number of successes (or defectives) from a finite population without replacement. It is characterized by the total number in the population, the number of successes in the population, the sample size, and the number of observed successes, and is ideal for situations where the draws are dependent due to the absence of replacement.

Combinatorics and Combination Formula

Key to understanding and computing probabilities in sampling without replacement is the use of combinations. The combination formula, often noted as 'n choose k', counts the number of ways to select a subset of items from a larger set regardless of order. This concept is used to determine both the total number of possible selections and the favorable selections that meet the event's criteria.

Probability Mass Function

In the context of discrete random variables like the number of defectives observed, the probability mass function (PMF) assigns a probability to each possible value of the variable. For a hypergeometric distribution, the PMF is given in terms of combination formulas that compute the likelihood of obtaining a specific number of successes under the conditions of finite sampling without replacement.

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