Paradoxes
Birthday paradox: among n people, $P(conflict)=1-P(\neg conflict)=1-(1)(\frac{364}{365})(\frac{363}{365})\dots\frac{365-n+1}{365}=1-\frac{365!}{(365-n)!365^n}$
Monty Hall
Base rate fallacy: thinking $$P(A|C)>P(B|C) \implies P(A)>P(B)
Gambler’s Fallacy aka Monte Carlo fallacy: belief that, if an event (whose occurrences are independent and identically distributed) has occurred more frequently than expected, it is less likely to happen again in the future (or vice versa)
Simpson’s paradox
Berkson’s paradox: subsets of data have different correlations.
E.g. talent uncorrelated with attractiveness, but sampling celebrities makes you think it’s negative.
E.g. good fries correlate with good burgers, but someone who missed restaurants with bad fries+burgers will think it’s negative.
You have 52 playing cards (26 red, 26 black). You draw cards one by one. A red card pays you a dollar. A black one fines you a dollar. You can stop any time you want. Cards are not returned to the deck after being drawn. What is the optimal stopping rule in terms of maximizing expected payoff?
Let’s say that you’re drawing \(N\) cards (without replacement) from a standard 52 card poker deck. Each card is unique and part of 4 different suits and 13 different ranks. Compute the probability that you will get a pair (two cards of the same rank) from a hand of \(N\) cards.