Probability and combinatorics help students count possible outcomes and measure how likely events are. This cheat sheet covers sample spaces, events, probability rules, permutations, combinations, and conditional probability. Students need these tools for games of chance, surveys, genetics, simulations, and many standardized test problems.
A clear reference makes it easier to decide which counting method or probability formula applies.
The core idea is that probability compares favorable outcomes to total outcomes, often written as . Combinatorics provides counting shortcuts such as , , and . Probability rules such as and help organize overlapping and related events.
Conditional probability uses to update probability when new information is known.
Key Facts
- The probability of an event is when all outcomes are equally likely.
- Every probability must satisfy , where means impossible and means certain.
- The complement rule is , so the probability that does not happen is found by subtracting from .
- The addition rule is , and for mutually exclusive events it becomes .
- For independent events, the multiplication rule is .
- The number of permutations of objects chosen from objects is when order matters.
- The number of combinations of objects chosen from objects is when order does not matter.
- Conditional probability is , where .
Vocabulary
- Sample Space
- The sample space is the set of all possible outcomes of an experiment.
- Event
- An event is a subset of the sample space that contains the outcomes being considered.
- Complement
- The complement of event , written , is the event that does not occur.
- Independent Events
- Independent events are events where the occurrence of one event does not change the probability of the other.
- Permutation
- A permutation is an arrangement in which order matters, counted by .
- Combination
- A combination is a selection in which order does not matter, counted by .
Common Mistakes to Avoid
- Using permutations when order does not matter is wrong because it counts the same group multiple times. Use for selections and for arrangements.
- Forgetting to subtract the overlap in is wrong because outcomes in get counted twice. Use .
- Treating dependent events as independent is wrong because the first event can change the probability of the second event. Use conditional probability or adjust the sample space after the first outcome.
- Using when choices cannot repeat is wrong because assumes each choice has the same number of options every time. For no repetition, use a decreasing product or .
- Confusing with is wrong because the given information is different. means event is already known to have happened.
Practice Questions
- 1 A bag has red marbles, blue marbles, and green marbles. What is the probability of choosing a blue marble?
- 2 How many different -letter arrangements can be made from distinct letters if no letter is repeated?
- 3 A committee of students is chosen from students. How many different committees are possible?
- 4 Explain how you can tell whether a problem should use a permutation or a combination without doing any calculation.
Understanding Probability & Combinatorics
A sample space is more than a list to memorize. It is a model of every result that the situation allows. A good model must state what counts as one outcome.
For two coin flips, the outcomes are ordered pairs because first flip and second flip are different positions. This matters when outcomes appear similar at first.
In a card problem, drawing a heart then a king is not the same description as drawing one card that is both a heart and a king. Before calculating, define the experiment carefully and check whether each listed outcome truly has the same chance.
Counting methods work because they prevent repeated listing. The main decision is whether changing the order creates a new result. Assigning gold, silver, and bronze medals uses order, since each placement has a different meaning.
Choosing three students for a committee does not use order, since the same three names form the same group. A useful check is to write one selection in two different orders. If those orders describe the same final result, use a combination.
Factorials are useful when arranging distinct objects, but zero factorial has the value one. This rule keeps counting patterns consistent when no objects remain to arrange.
Tree diagrams show how multistep events are built. Each branch represents a possible next result, and the probabilities along one complete path are multiplied. The probabilities of separate paths that satisfy the same condition are added.
Trees are especially useful when draws happen without replacement. For example, the chance of selecting two red marbles changes after the first red marble is removed. The second branch must use the new total and the new number of red marbles.
This is different from replacement, where the original mix is restored. Students often multiply probabilities automatically without noticing that the first result changed the second chance.
Conditional probability describes a restricted group rather than the whole population. If a survey says a person is a cyclist, the relevant comparison group becomes cyclists only. This can produce a result very different from the overall survey result.
Tables with rows and columns help keep the restricted total visible. Read the condition first, then find the matching row or column, and use that subtotal as the new whole. Be careful with words such as at least, exactly, either, both, and given.
At least includes several cases. Either can include both unless the wording says otherwise.
A final reasonableness check helps. Probabilities should match the story, and a more restricted event should not be more likely than the larger event containing it.