Probability

Builds on

Required prerequisites, all the way down: 6

You will learn to

• List sample spaces for simple experiments, using tables or tree diagrams for two stages.
• Compute probabilities of events with equally likely outcomes as fractions.
• Compare theoretical probabilities with relative frequencies from repeated trials or simulations.

Definition

• A sample space is the set of possible outcomes of an experiment; an event is a subset of the sample space.
• If all outcomes in a finite sample space S are equally likely, the probability of an event A is P(A) = |A|/|S|.
• The relative frequency of an event in n trials is the number of times it occurred divided by n.

Theorem

• 0 ≤ P(A) ≤ 1, P(S) = 1, and P(not A) = 1 − P(A).
• As the number of independent trials grows, relative frequencies tend to settle near the probability (the law of large numbers, stated informally).

Common misconceptions

• The gambler's fallacy: thinking a fair coin is 'due' to land heads after several tails.
• Assuming all outcomes are equally likely, for example all sums of two dice.
• Believing a probability of 1/6 means the event happens exactly once in every 6 trials.

Leads to

Optional discoveries

• Measure-theoretic probability Kolmogorov (1933) placed probability on the foundation of measure theory.
• ← Ratios, rates and proportional reasoning
• ← Counting principles

Learning pathways

• Conventional K–12 progression
• Adult beginner pathway