A shelf in the Metro Department Store contains 80 colored...

A shelf in the Metro Department Store contains 80 colored ink cartridges for a popular ink-jet printer. Six of the cartridges are defective. If a customer selects 2 cartridges at random from the shelf, what is the probability that
a. Both are defective?
b. At least 1 is defective?

Key Concepts

Combinatorial Analysis

This concept involves counting the number of ways to choose items from a set, typically using combinations. It is crucial when order does not matter, as in selecting multiple items from a shelf. The combination formula is used to calculate both total possible outcomes and favorable outcomes in probability problems.

Sampling Without Replacement

This concept refers to the process of selecting items from a finite population where each selected item is not returned to the population before the next selection. This changes the probabilities of subsequent selections, leading to dependent events, and is a common scenario in problems involving selections from a limited set.

Hypergeometric Distribution

The hypergeometric distribution models the probability of a given number of successes in draws from a finite population without replacement. It is especially relevant when dealing with scenarios involving defective versus non-defective items in a fixed group, enabling exact probability calculations for events like drawing defective items.

Complement Rule

The complement rule in probability is used to calculate the probability of an event by subtracting the probability of its complement from one. It simplifies the process of finding probabilities for events such as 'at least one defect' by instead calculating the probability of 'no defects' and then subtracting from one.

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