Independence

Two events are independent when the occurrence of one provides no information about the other: knowing the result of the first does not change the probability of the second in any way. Conversely, two events are dependent if the result of one changes the chances of the other. Independence is one of the most useful, and most misunderstood, notions in the study of randomness.

The most accurate mental image is that of an absence of memory. A fair coin does not remember its previous flips: it has no way of knowing that it has just landed on heads several times, and therefore has no reason to correct course on the next flip.

Formally, independence is checked on the probabilities. Two events are independent when the probability that they both occur equals the product of their probabilities. For two coin flips, the probability of getting heads then heads is 0.5 multiplied by 0.5, that is 0.25, in other words 1 chance in 4.

The most concrete distinction contrasts drawing with replacement and without replacement. When you draw a card, put it back in the deck, and reshuffle, each draw is independent. If you do not put the card back, the deck changes: drawing an ace goes from 4 in 52 on the first draw to 3 in 51 on the next, which makes the draws dependent.

The most widespread trap is the gambler's fallacy, which consists of believing that after five heads in a row, tails becomes due. When the flips are independent, the probability stays at 0.5 on the sixth flip just as on the first; the past series exerts no pull on the future.

This property is guaranteed by the site's repeated tools. Every die roll, every coin flip, every call to a number generator starts fresh, without taking earlier results into account, which preserves fairness draw after draw.