Probability & Combinatorics Cheat Sheet

A printable reference covering sample spaces, probability rules, permutations, combinations, factorials, and conditional probability for grades 9-11.

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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 $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$ . Combinatorics provides counting shortcuts such as $n!$ , $_nP_r=\frac{n!}{(n-r)!}$ , and $_nC_r=\frac{n!}{r!(n-r)!}$ . Probability rules such as $P(A^c)=1-P(A)$ and $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ help organize overlapping and related events. Conditional probability uses $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ to update probability when new information is known.

Key Facts

The probability of an event is $P(A)=\frac{\text{number of outcomes in }A}{\text{number of outcomes in the sample space}}$ when all outcomes are equally likely.
Every probability must satisfy $0\le P(A)\le 1$ , where $0$ means impossible and $1$ means certain.
The complement rule is $P(A^c)=1-P(A)$ , so the probability that $A$ does not happen is found by subtracting from $1$ .
The addition rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ , and for mutually exclusive events it becomes $P(A\cup B)=P(A)+P(B)$ .
For independent events, the multiplication rule is $P(A\cap B)=P(A)P(B)$ .
The number of permutations of $r$ objects chosen from $n$ objects is $_nP_r=\frac{n!}{(n-r)!}$ when order matters.
The number of combinations of $r$ objects chosen from $n$ objects is $_nC_r=\frac{n!}{r!(n-r)!}$ when order does not matter.
Conditional probability is $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ , where $P(B)\ne 0$ .

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 $A$ , written $A^c$ , is the event that $A$ 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 $_nP_r=\frac{n!}{(n-r)!}$ .
Combination: A combination is a selection in which order does not matter, counted by $_nC_r=\frac{n!}{r!(n-r)!}$ .

Common Mistakes to Avoid

Using permutations when order does not matter is wrong because it counts the same group multiple times. Use $_nC_r$ for selections and $_nP_r$ for arrangements.
Forgetting to subtract the overlap in $P(A\cup B)$ is wrong because outcomes in $A\cap B$ get counted twice. Use $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ .
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 $n^r$ when choices cannot repeat is wrong because $n^r$ assumes each choice has the same number of options every time. For no repetition, use a decreasing product or $_nP_r$ .
Confusing $P(A\mid B)$ with $P(B\mid A)$ is wrong because the given information is different. $P(A\mid B)$ means event $B$ is already known to have happened.

Practice Questions

1 A bag has $5$ red marbles, $3$ blue marbles, and $2$ green marbles. What is the probability of choosing a blue marble?
2 How many different $4$ -letter arrangements can be made from $8$ distinct letters if no letter is repeated?
3 A committee of $3$ students is chosen from $10$ 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.