Fairness: When Outcomes Are Not Equally Likely
Fairness: When Outcomes Are Not Equally Likely is a Grade 7 math topic in Big Ideas Math Advanced 2, Chapter 15: Probability and Statistics. Students learn that equal probability must be justified by experimental evidence or knowledge of the situation, and cannot be assumed simply because multiple outcomes exist. When outcomes are not equally likely, the standard theoretical probability formula does not apply without additional data.
Key Concepts
Not all outcomes are equally likely just because they exist. Equal probability requires either experimental evidence or knowledge that outcomes have the same chance of occurring. When outcomes are not equally likely, theoretical probability cannot be calculated as $P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$.
Common Questions
Why can you not always assume outcomes are equally likely?
Just because multiple outcomes exist does not mean they occur with equal frequency. For example, a weighted coin or a bag with unequal numbers of marbles produces outcomes with different probabilities.
When can you assume equally likely outcomes?
You can assume equally likely outcomes when you know the physical setup guarantees it (like a fair coin or a standard number cube) or when experimental data confirms equal frequencies.
How do you find probability when outcomes are not equally likely?
You need experimental evidence or additional information about the actual frequencies. For example, a bag with 3 red and 1 blue marble: P(red) = 3/4, not 1/2, because the outcomes are not equally likely.
Is a game fair if the outcomes are not equally likely?
Not necessarily. A game is fair if all players have equal probability of winning. If outcomes favor one player over another, the game is unfair.