St. Petersburg paradox: a gambling game wherein the payoff increases exponentially with each coin flip has an infinite exp[ected value --so what’s a reasonable stake?
Random chord problem: The probablity of a random chord on a circle being longer than the side of an inscribed square has two values, depending on whether “uniformly distributed” is defined in terms of angle or distance from the center.
Buffon’s Needle: involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page. The complexity of the analysis depends on the length of the needle compared to the distance between the lines.
Banach matchbox: A man keeps two matchboxes, one in each pocket. Each box contains exactly n matches. Whenever he wants a match he reaches at random into one of his pockets. When he finds that the box he picks is empty, what is the probability that the other one has exactly k matches (k = 0, 1, 2,…n)?
Hot hand: an analysis demonstrating that apparent streaks (coinflipping, sinking basketball free throws) are predictable distributions of binomial events.
Gambler’s fallacy: thinking that you can beat the odds by either selecting numbers that have not been chosen in recent drawings, or by selecting numbers that have come up more frequently than expected in recent drawings; a common error in lotteries and roulette.
Gambler’s ruin: In theory, if you are playing a game where the odds are in your favor, and you play long enough, eventually you will come out a winner. In practice, though, you have a finite stake, and you are going to quit if you run out of money. Thus you could very possibly walk away a loser from a game that is actually in your favor.
Drunkard’s walk: a distribution around a starting point traced by a randomly moving particle (or a drunk taking single steps around a lamp post).
Monty Hall: Guess that a prize is behind one of three doors - Monty then opens one of the unpicked doors. Should you switch doors?