Probability Theory · Mathematics

Counting principles

The multiplication principle: if one task can be done in m ways and another in n ways, both can be done in m × n ways. A permutation is an ordered arrangement, and a combination is a selection in which order does not matter.

n! = n × (n − 1) × … × 1. For example ⁵C₂ = 10.

Worked exampleCombinations

In how many ways can a committee of 3 be chosen from 8 people?

Order does not matter.

Answer: 56 ways.

Independent and dependent events

Events are independent if one does not change the probability of the other: P(A ∩ B) = P(A) × P(B). For dependent events, P(A ∩ B) = P(A) × P(B | A), where P(B | A) is the conditional probability of B given that A has happened.

Conditional probability and the addition rule.

Worked exampleDrawing without replacement

A bag has 4 red and 6 blue balls. Two are drawn without replacement. Find P(both red).

First red.
Second red, given the first.
Multiply.

Answer: 2/15.

Probability distributions

A binomial distribution counts the number of successes in n independent trials, each with the probability p of success: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ with q = 1 − p. The mean is np and the variance is npq.

P(X = r) for the number of heads r = 0 to 4.

Try itBinomial probability

Choose the number of trials n (up to 3 successes shown), the probability p, and r (0 to 3).

Check yourself ⁶C₂ =

Check yourself The number of ways to arrange 4 different books on a shelf is

Check yourself If P(A) = 0.5, P(B) = 0.4 and A, B are independent, P(A ∩ B) =

Key terms

Permutation: An ordered arrangement.
Combination: A selection where order does not matter.
Conditional probability: The probability of B given that A has happened.
Binomial distribution: The distribution of successes in n independent trials.

Summary

ⁿPᵣ = n!/(n − r)! and ⁿCᵣ = n!/(r!(n − r)!).
Independent events: multiply; dependent events use conditional probability.
Binomial: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ.
The binomial mean is np and the variance is npq.