🟢 Stats: Classic Probability Puzzles

Monty Hall You pick Door 1 (1/3 chance of winning). Host opens Door 3 (goat). Should you switch to Door 2?

Always switch. Switching = 2/3. Staying = 1/3.

Your initial 1/3 didn't change. The other two doors had 2/3 combined. Host revealed one was wrong, so the remaining door inherits the full 2/3.

Birthday Problem How many people for >50% chance of a shared birthday? 23.

Compute complement: P(no match) = (364/365)(363/365)...(343/365). At n=23, P(no match) < 0.5.

Expected value of a die roll E = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5

"Would you pay $4 to play a game where you win whatever you roll in dollars?" No — expected payout is $3.50, you'd lose $0.50 on average.