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ProbabilityHigh school3 topic explainers

Probability course

Counting principles through expected value.

Topics in this course

  1. 1.Counting

    Counting principles answer how many ways something can happen without listing every case. The multiplication principle says that if one choice has mm options and a second has nn, there are mnmn combined outcomes. Factorials, n!=n(n−1)⋯1n! = n(n - 1)\cdots 1, count the ways to arrange nn distinct objects in a row.

    • Multiplication principle for sequences of independent choices.
    • Order matters: permutation; order does not matter: combination.
    • C(n,r)=C(n,n−r)C(n, r) = C(n, n - r).
    Quick check: How many ways can 5 books be arranged on a shelf?

    120120

    Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study counting

  2. 2.Probability Rules

    Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain). When outcomes are equally likely, P(A)=favourable outcomestotal outcomesP(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}. The complement rule P(not A)=1−P(A)P(\text{not } A) = 1 - P(A) is often the fastest route, especially for at-least-one questions.

    • Mutually exclusive events cannot happen together, so P(A∩B)=0P(A \cap B) = 0.
    • Independent means P(B∣A)=P(B)P(B \mid A) = P(B); it is not the same as mutually exclusive.
    • At least one: 1−P(none)1 - P(\text{none}).
    Quick check: Two coins are tossed. What is the probability of two heads?

    14\frac{1}{4}

    Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study probability rules

  3. 3.Expected Value

    The expected value of a random variable is its long-run average outcome: each possible value weighted by its probability, E(X)=∑xipiE(X) = \sum x_i p_i. It need not be a value the variable can actually take. A game with expected value −$0.50-\$0.50 loses fifty cents per play on average over many plays, even though no single play loses exactly fifty cents.

    • E(X)=∑xiP(xi)E(X) = \sum x_i P(x_i); check the probabilities sum to 1.
    • Linearity: E(aX+b)=aE(X)+bE(aX + b) = aE(X) + b and E(X+Y)=E(X)+E(Y)E(X + Y) = E(X) + E(Y).
    • A fair game has expected value 0.
    Quick check: A fair die: find the expected value of one roll.

    3.53.5

    Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study expected value

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