GED · Mathematical Reasoning · 6 min read
Probability and Counting Possible Outcomes
- Calculate theoretical and experimental probabilities, complements, and compound events using multiplication and addition rules.
- Determine total possible outcomes using the Fundamental Counting Principle, tree diagrams, systematic lists, and two-way frequency tables.
- Distinguish between permutations and combinations and solve arrangement and grouping problems using the TI-30XS calculator.
Foundations of Probability and Sample Spaces
Probability measures the likelihood that a specific event will occur. When evaluating any situation involving chance, the probability of any event is always a value between 0 and 1, inclusive. A probability of 0 means the event is impossible and cannot occur, while a probability of 1 means the event is certain to occur. In mathematical problems, probabilities can be represented as fractions, decimals, or percentages. Unless directions say otherwise, simplify fractional probability answers to lowest terms.
The sample space is the complete set of all possible outcomes of an experiment. When all outcomes in a sample space are equally likely, you find the probability by dividing the number of favorable outcomes by the total number of possible outcomes. The sum of the probabilities of all distinct possible outcomes in a sample space always equals 1. For example, a fair coin has two equally likely outcomes, heads and tails, each with a probability of 1/2. Similarly, a fair six-sided die has the sample space {1, 2, 3, 4, 5, 6}, with each number having a probability of 1/6. Another standard model is a standard deck of playing cards, which contains 52 cards divided equally into four suits of 13 cards each.
It is also important to distinguish how probabilities are determined. Theoretical probability comes from a mathematical model of an experiment; when outcomes are equally likely, you divide favorable outcomes by total outcomes. In contrast, experimental probability is calculated from the actual results observed during repeated trials of an experiment.
Every event has an opposite called its complement. The complement of event A is the event that A does not occur, often phrased as 'not A'. Because the total probability of all outcomes in a sample space is 1, the probability of the complement of an event is calculated as 1 minus the probability of the event: P(not A) = 1 - P(A). For example, if you know the theoretical probability of rolling a specific number on a fair six-sided die is 1/6, the probability of not rolling that number is 1 - 1/6 = 5/6.
Common mistakes
- Double-counting outcomes when finding the probability of overlapping events by forgetting to subtract P(A and B).
- Treating dependent events as independent by failing to reduce the total number of possible outcomes when drawing without replacement.
- Using a permutation instead of a combination when the order of selection does not matter, such as selecting an unordered committee.
- Dividing by the grand total instead of the row or column total when calculating a conditional probability from a two-way frequency table.
Key points
- Probability is always a value between 0 and 1 inclusive; 0 means impossible, 1 means certain, and the sum of all outcomes in a sample space equals 1.
- The complement of event A is 'not A', with probability calculated as P(not A) = 1 - P(A).
- For independent events, P(A and B) = P(A) * P(B); for dependent events, P(A and B) = P(A) * P(B given A).
- For mutually exclusive events, P(A or B) = P(A) + P(B); for overlapping events, P(A or B) = P(A) + P(B) - P(A and B) to avoid double-counting.
- The Fundamental Counting Principle multiplies choices across stages, while tree diagrams, systematic lists, and grids systematically display sample spaces.
- In two-way tables, overall probabilities divide by the grand total, whereas conditional probabilities divide by the total for the given condition.
- Order matters in permutations (like selecting specific officers), but order does not matter in combinations (like forming a committee); both can be computed on the TI-30XS via the [prb] menu.
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