Pi appears in non-circular formulas mainly because of hidden rotational symmetry and because oscillations are mathematically tied to circular motion. Some appearances come from how constants are defined.
Starter thread · AI-reviewed against its sources
In a random group of just 23 people, the chance that at least two share a birthday is slightly over 50 percent. It seems wrong, but it is true.
Starter thread · AI-reviewed against its sources
Simpson's paradox is when a trend seen in several groups disappears or reverses once the groups are combined, usually because a hidden confounding variable is unevenly spread across them.
Starter thread · AI-reviewed against its sources
Most people assume the two closed doors are a 50-50 choice, but switching wins two thirds of the time. Researchers point to missed host constraints, equal-odds thinking and emotional biases.
Starter thread · AI-reviewed against its sources
A 52-card deck has about 8×10^67 possible orders, far more than all shuffles in history, so a properly randomized order has almost certainly never occurred before.
Starter thread · AI-reviewed against its sources
The gambler's fallacy is the false belief that past outcomes of independent random events change the odds of future ones, so a streak makes the opposite result "due."
Starter thread · AI-reviewed against its sources
A coin game has an infinite expected payoff, yet almost no one would pay much to play it. The puzzle challenges the idea that rational choice means maximizing expected value.
Starter thread · AI-reviewed against its sources
