Probability is area in a square of area 1. Each axiom is a statement about areas.
Three disjoint pieces that fill the whole line. Let them predict first.
Reveal: the three probabilities add to 1, so P(W=5) = 1 - a - b.
Paint A, paint B, then notice the lens got two coats. Remove one coat.
Build the chain one link at a time: each region sits inside the next, then the sum overshoots.
Two rods that are too long for the unit interval. Let them predict first.
Reveal: push A to the left wall and B to the right wall; the forced overlap is 0.70 + 0.55 - 1 = 0.25.
Two rays on the number line that share exactly one point. Let them predict first.
Reveal: the overlap is the single point W = 2 and the union is Omega, so 1 = c + d - P(W=2) and P(W=2) = c + d - 1.
Two intervals in the unit interval. Let them slide B in their head and commit to a smallest and a largest overlap.
Reveal: slide B across A. The overlap runs from 0 up to 0.35, then the second case with the complement of A.
Failures can overlap or be disjoint. The union bound handles the worst case.
Two different notions: one is about sets, one is about a product. Let them decide before the reveal.
Reveal: 0.1575 is in the range [0, 0.35], so independence is possible; being exclusive as well is impossible for these numbers.
Compare what happens to A inside B in two cases.
Four rectangles, each the product of its sides. The table on the right is the same grid.
Three pairs of events. Let them decide which pairs could be mutually exclusive.
Reveal: (a) yes, 2/9 + 7/9 = 1 exactly tiles Omega; (b) no, 14/9 > 1 and the overlap is at least 5/9; (c) yes, 4/9 <= 1.
Three questions about the same two probabilities. Let them answer all three.
Reveal: (d) yes, 14/81 is inside [0, 2/9]; (e) no, 14/81 is not 0; (f) the minimum is 0.
Two events cut the sample space into four pieces. Name each piece with a complement sign where needed.
Each condition rules out some atoms. Let them name which atoms before the reveal.
Reveal: A intersect B-complement and A-complement intersect B are empty, so A = B.
Seven take-aways, then the drills.