Two balls are chosen randomly from an urn containing 8...

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $\$ 2$ for each black ball selected and we lose $\$ 1$ for each white ball selected. Let $X$ denote our winnings. What are the possible values of $X,$ and what are the probabilities associated with each value?

Key Concepts

Outcome Transformation in Random Processes

This concept involves mapping the results of the random experiment (e.g., counts of different types of balls drawn) to a numerical payoff. It is a key step in many probability problems where the interest lies not just in the occurrence of an event, but in a function of the event that quantifies gains or losses.

Combinatorial Analysis

This concept involves using combinations to count the number of ways events can occur. In problems with selection from distinct groups, combinatorial analysis is crucial for finding the probabilities by calculating the number of favorable outcomes divided by the total number of possible selections.

Discrete Random Variables

A discrete random variable is one that can take on only a countable number of distinct values. In this context, the winnings are defined as the random variable whose possible outcomes and associated probabilities are derived from the underlying random experiment.

Random Sampling Without Replacement

This concept refers to the method of drawing items from a population where each selection changes the composition of the population because the items selected are not returned to the pool. It is important in probability because the lack of replacement creates dependencies between draws, affecting the probability calculations.

Hypergeometric Distribution

The hypergeometric distribution models the probability of k successes (e.g., drawing a ball of a particular type) in n draws from a finite population without replacement. It is relevant in determining the likelihood of various outcomes when items are selected from different categories.

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