Probability

Two games that each guarantee a loss can become a winning strategy when played alternately. One game's result influences the other's conditions, allowing you to selectively hit favorable outcomes. This is known as 'Parrondo's paradox.'

The 'Sleeping Beauty problem' is a famous probability paradox. If a coin lands heads, she is woken once; if tails, twice with memory erased. The 'Halfer' camp says heads is 1/2, the 'Thirder' camp says 1/3, and both answers are logically valid.

To perfectly randomize a deck of cards, a riffle shuffle needs just 7 repetitions, while the overhand (Hindu) shuffle requires about 10,000.

If a disaster has a 0.1% chance and a 99%-accurate alarm is installed, out of ~110 alarm days in 10,000, about 100 are false alarms. Over 90% are wrong—yet dismissing the alarm means ignoring a 99%-accurate warning. This is the "base rate fallacy," where probability defies intuition.