Core Probability Rules

Complements and unions

The complement rule is useful when an event is hard to count directly. Instead of counting at least one success, you can count zero successes and subtract from 1.

For two events, adding P(A) and P(B) counts overlap twice. The addition rule corrects this by subtracting P(A and B).

Intersections and independence

If events are independent, the occurrence of one event does not change the probability of the other. Then P(A and B) = P(A)*P(B).

If events are not independent, use conditional probability: P(A and B) = P(A)*P(B|A). Confusing these two cases is a common source of errors.

• Mutually exclusive events have P(A and B) = 0.
• Mutually exclusive does not imply independent unless one event has probability 0.
• Draw a simple Venn diagram to track overlap.

Compute probability of passing at least one quiz

A student has probability 0.7 of passing Quiz 1 and 0.8 of passing Quiz 2, assume independence.

1. Define A = pass Quiz 1 and B = pass Quiz 2.
2. Use complement: P(at least one pass) = 1 - P(fail both).
3. Compute fail probabilities: P(A^c)=0.3 and P(B^c)=0.2.
4. Multiply for independent failures: P(fail both)=0.3*0.2=0.06.

Common mistakes

• Adding probabilities of overlapping events without subtracting intersection.
• Multiplying probabilities without checking independence.
• Using mutually exclusive and independent as if they mean the same thing.
• Forgetting that all probabilities must stay between 0 and 1.

Try one

Sources

• OpenStax: Introductory StatisticsOpen textbook covering descriptive statistics, probability, inference, and regression fundamentals.
• NIST/SEMATECH e-Handbook of Statistical MethodsAuthoritative reference on probability models, inference procedures, and practical statistical diagnostics.