Probability: Courses, Topics, Worked Examples and Practice

About Probability

Probability measures how likely events are, on a scale from 0 (impossible) to 1 (certain). It starts with counting equally likely outcomes and grows into conditional probability, independence, random variables, expected value and the standard distributions such as the binomial and the normal.

Intuition is often wrong here, which is what makes the subject valuable. The chance of at least one six in four rolls is $1 - (5/6)^4 \approx 0.52$ , not $4/6$ , and a positive result from an accurate test can still be more likely false than true when the condition is rare. Careful counting and Bayes' rule correct those instincts.

Probability is used in insurance, medicine, weather forecasting, games, quality control and machine learning. Any decision made under uncertainty, from setting a premium to judging a test result, depends on these ideas, and statistics courses assume them from the first week.

Probability

Counting principles through expected value. (High school)

Counting

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

Probability Rules

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

Expected Value

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

Full Probability course outline

How to study Probability

Write the sample space or a tree diagram before computing anything.
Use the complement for "at least one" questions: $P(\text{at least one}) = 1 - P(\text{none})$ .
Decide whether order matters before choosing permutations or combinations.
Check that probabilities in a distribution add to 1 and that expected values are plausible.

Before you start: Fractions, algebra and comfort with counting; the multiplication principle and factorials are used from the first lesson.
Where it leads: Statistics, where probability models are used to draw conclusions from data, and later machine learning, actuarial science and quantitative finance.

Key terms in Probability

Sample space: The set of every possible outcome of an experiment. Rolling two dice has 36 equally likely ordered outcomes.
Independent events: Events where one happening does not change the chance of the other, so $P(A \cap B) = P(A)P(B)$ .
Conditional probability: The chance of $A$ given that $B$ happened: $P(A \mid B) = P(A \cap B)/P(B)$ , read from the restricted sample space.
Expected value: The long-run average of a random variable, $E[X] = \sum x\,P(X = x)$ ; it need not be a value $X$ can actually take.
Complement: Everything outside an event. $P(\text{not } A) = 1 - P(A)$ , often the fastest route to "at least one" questions.
Binomial distribution: The count of successes in $n$ independent trials with the same success chance $p$ , with $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ .