Probability Rules & Conditional Probability Cheat Sheet

A printable reference covering addition rules, complements, conditional probability, independence, multiplication rules, and Bayes’ theorem for grades 9-12.

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Probability rules help students describe chance in a clear, organized way. This cheat sheet covers the formulas used to combine events, find complements, handle overlapping events, and solve conditional probability problems. It is useful for homework, tests, data analysis, and interpreting real-world situations involving risk or uncertainty. The most important ideas are that probabilities range from $0$ to $1$ , complements add to $1$ , and overlapping events must be counted carefully. Conditional probability measures the chance that one event happens when another event is already known to have happened. Independence, multiplication rules, tree diagrams, and Bayes’ theorem help students connect multi-step probability situations.

Key Facts

Every probability satisfies $0 \le P(A) \le 1$ , where $P(A)=0$ means impossible and $P(A)=1$ means certain.
The complement rule is $P(A^c)=1-P(A)$ , where $A^c$ means the event $A$ does not occur.
The general addition rule is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ .
If events $A$ and $B$ are mutually exclusive, then $P(A\cap B)=0$ and $P(A\cup B)=P(A)+P(B)$ .
Conditional probability is $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$ , as long as $P(B)>0$ .
The general multiplication rule is $P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$ .
Events $A$ and $B$ are independent if $P(A\mid B)=P(A)$ , which is equivalent to $P(A\cap B)=P(A)P(B)$ .
Bayes’ theorem is $P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$ , where $P(B)>0$ .

Vocabulary

Event: An event is a set of outcomes from a probability experiment.
Sample Space: The sample space is the set of all possible outcomes of an experiment.
Complement: The complement of event $A$ , written $A^c$ , is the event that $A$ does not happen.
Union: The union $A\cup B$ is the event that $A$ happens, $B$ happens, or both happen.
Intersection: The intersection $A\cap B$ is the event that both $A$ and $B$ happen.
Conditional Probability: Conditional probability $P(A\mid B)$ is the probability that $A$ occurs given that $B$ has already occurred.

Common Mistakes to Avoid

Adding overlapping events without subtracting the intersection is wrong because outcomes in $A\cap B$ get counted twice in $P(A)+P(B)$ .
Treating mutually exclusive events as independent is wrong because if $A$ and $B$ cannot both happen, knowing one occurred changes the probability of the other.
Reversing conditional probabilities is wrong because $P(A\mid B)$ and $P(B\mid A)$ usually describe different situations and are not usually equal.
Using $P(A\cap B)=P(A)P(B)$ without checking independence is wrong because that formula only works when $A$ and $B$ are independent.
Forgetting that probabilities must stay between $0$ and $1$ is wrong because answers such as $1.2$ or $-0.1$ cannot represent valid probabilities.

Practice Questions

1 If $P(A)=0.45$ , $P(B)=0.30$ , and $P(A\cap B)=0.12$ , find $P(A\cup B)$ .
2 A class has $18$ students who play soccer, $12$ who play basketball, and $7$ who play both. If one student is chosen at random, what is the probability the student plays soccer or basketball out of $30$ students?
3 If $P(A)=0.60$ and $P(B\mid A)=0.25$ , find $P(A\cap B)$ .
4 Explain how you can tell from a two-way table whether two events are independent without relying only on the words in the problem.

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