When Independence Fails

Not all events are independent. If you draw a card from a deck without putting it back, the odds for the next card change. Drawing an ace and then another ace is a dependent event — the first draw changes what is left. Confusing dependent and independent events is as costly as the gambler's fallacy itself.

Conditional probability — the chance of B given A — handles these cases. After drawing an ace, a second ace is 3 in 51; if the first card was not an ace, it is 4 in 51. This is the math behind card counting in blackjack, where tracking which cards remain shifts the odds in the player's favor.