Objectives
Explain the formula, show each numerical step, and state its assumptions.
Mathematical reasoning
The sample space S contains possible outcomes and an event A is a subset. P(A) lies between 0 and 1. Disjoint events cannot occur together; independent events do not affect one another's chances. These properties differ.
The addition rule P(A∪B)=P(A)+P(B)−P(A∩B) avoids double-counting. The multiplication rule is P(A∩B)=P(A)P(B|A); under independence this becomes P(A)P(B).
Choosing k from n objects without order yields C(n,k)=n!/[k!(n−k)!]. When roles or order matter, use n!/(n−k)!. Match the counting method to the question.
Key formula
C(n,k)=n! / [k! (n−k)!] P(A∪B)=P(A)+P(B)−P(A∩B)
Worked example: step by step
- Select two unordered members from five: C(5,2)=10.
- Chair and secretary are ordered roles, giving 5×4=20 outcomes.
- Fair die: A=even={2,4,6}, B=greater than 4={5,6}. P(A∪B)=3/6+2/6−1/6=4/6.
Independent exercises
- How many unordered committees of three can be chosen from six?
- Find P(even or greater than 3) for a fair die.
- What is the chance of three heads in three independent fair-coin flips?
Optional base R check
choose(5,2)
choose(6,3)
outcomes <- 1:6
mean(outcomes %% 2 == 0 | outcomes > 4)