Big Bored Securities has two brands of personal computers in...

Big Bored Securities has two brands of personal computers in the Information Center to use for demonstrations, brand $y$ and brand $z$. If two personal computers are selected at random, the probability that both are brand $y$ is $1 / 2$. What is the smallest number of personal computers that could be in the Information Center?

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Key Concepts

Combinatorial Probability

Combinatorial probability involves determining the likelihood of an event by taking the ratio of favorable outcomes to the total number of outcomes, often using combinations to count these possibilities. This method is essential in discrete probability problems where the outcome space can be enumerated using combinatorial formulas.

Combination Formula

The combination formula, expressed as C(n, k) = n!/(k!(n-k)!), is a fundamental tool in combinatorics used to count the number of ways to choose k items from a collection of n items where order does not matter. This concept is key when calculating probabilities in scenarios that involve selection from a set without regard to sequence.

Diophantine Equations in Probability

In many probability problems involving discrete objects, the equations derived from the probability conditions require integer solutions, leading to Diophantine equations. Solving such equations involves finding integer values that satisfy the conditions of the problem, which is crucial when determining the smallest or optimal number of items that meet the criteria.

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