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M06.1 · Sample spaces, events and axioms

A free, illustrated module of 4 lessons and one interactive in M06 Probability I: Experiments, outcomes and sample spaces; Events as sets; The three axioms of probability; Rules that follow from the axioms.

M06.1 L1 · Core 4 lessons · about 100 min Live · in review

Part of M06 Probability I in the Mathematics track.

Every probability starts with a list of what could happen. These four lessons build that list, the sample space, for coins, dice and spinners; pick out events as sets of outcomes and combine them with or, and and not; give each event a number under three short axioms, drawn as areas on a square; and derive from those axioms alone the rules used everywhere after: the complement rule, the addition rule and the union bound, all worked on the grid of two dice.

What you need first. Sets, unions, intersections and complements (M01.6), the multiplication rule of counting (M01.8) and the geometric sum (M01.7). A little Python for the code sections: lists, sets, loops and functions (P01), each line of which is explained as it appears.

Lessons

1. Experiments, outcomes and sample spaces M06.1.1 25 min

   Before anything can have a probability, you must list what could happen. Learn to name an experiment, its outcomes and its sample space, to list them for coins and dice, and to tell finite, countably infinite and continuous sample spaces apart.

2. Events as sets M06.1.2 25 min

   An event is a set of outcomes: “the sum is 7” is six cells of the dice grid. Learn to read or, and and not as union, intersection and complement, to prove De Morgan’s laws by following one outcome, and to turn phrases like at least one into sets.

3. The three axioms of probability M06.1.3 25 min

   A probability gives every event a number under three rules: never negative, one for the whole sample space, and adding up over events that cannot happen together. See them as areas, use them on a loaded die and on flips until the first head, and meet two readings of the same number.

4. Rules that follow from the axioms M06.1.4 25 min

   From the three axioms alone, derive the complement rule, P(empty set) = 0, the rule for an event inside another, the addition rule for overlapping events and the union bound, each with its area picture, then check them on the dice grid with fractions and a simulation.

The interactive

M06.1-X1 · Event explorer. The 36 outcomes of two dice as a grid. Choose events A and B from presets or by clicking cells, then light up their union, intersection or a complement; the read-out counts the cells and checks that P(A or B) = P(A) + P(B) - P(A and B). It lives in Rules that follow from the axioms.