What independence means
Two events are independent when learning that one happened tells you nothing about the other. Two rolls of a fair die are the classic case: the first roll does not affect the second. Two events are dependent when the outcome of one changes the chance of the other.
In symbols, events A and B are independent if the chance of A given B equals the plain chance of A. This ties directly to conditional probability: if knowing B leaves the chance of A unchanged, they are independent. If knowing B shifts it up or down, they are dependent.
For independent events the rule for “both happen” is simply multiplication: P(A and B) = P(A) × P(B). For dependent events the multiplication still works, but the second factor must be the conditional chance: P(A and B) = P(A) × P(B | A).
Independent: two sixes in a row
A fair die has six sides, so the chance of a six is 1/6 each time. The die has no memory, so the chance of a six on the second roll is still 1/6 whatever happened on the first. Two sixes in a row:
1/6 × 1/6 = 1/36
The same reasoning covers coin tosses, roulette spins and shuffles of a freshly shuffled deck from one round to the next. The belief that a long run of one result makes the opposite “due” is the gambler’s fallacy, and it comes from treating independent events as if they were dependent.
Dependent: drawing without replacement
Now draw two cards from a standard 52-card deck, without putting the first back, and ask for two aces.
- First card is an ace: 4/52, since there are four aces.
- If that happened, 51 cards remain, 3 of them aces. Second card is an ace: 3/51.
Both: 4/52 × 3/51 = 12/2,652 = 1/221.
The second factor is 3/51, not 4/52, because the first draw changed what remains in the deck. That change is exactly what makes the events dependent.
The same question with replacement
Now put the first card back and reshuffle before drawing again. Each draw is the same 4/52 chance, and the draws are independent:
4/52 × 4/52 = 16/2,704 = 1/169
So “two aces” is 1 in 221 without replacement and 1 in 169 with replacement. The difference is small here because the deck is large, but it is real, and it grows as the pool gets smaller.
A small pool makes the gap obvious
Take a bag with 5 red marbles and 5 blue marbles, and draw two. The chance both are red:
- Without replacement: 5/10 × 4/9 = 20/90 = 2/9, about 22.2%.
- With replacement: 5/10 × 5/10 = 1/4, or 25%.
With only ten marbles, taking one out visibly changes the mix. Taking out a red leaves 4 red among 9, so another red is less likely than it was. A rough rule of thumb: the smaller the pool relative to the number you draw, the more dependence matters. Draw a few cards from a huge collection and the two answers are nearly the same; draw several from a small one and they diverge.
Dependence is not the same as cause
Dependent events are not necessarily linked by cause and effect. In the card example, the first draw does not “cause” anything to happen to the second card; it simply removes information-carrying cards from the pool. And two events can be dependent because they share a hidden common factor. The distinction between association and cause is covered in the article on correlation versus causation.
Dependence is what lets you learn
Dependence is also what makes updating possible. If hidden information is linked to something you can see, then seeing it should change your estimate. In a card game, the cards you can see are no longer in the unseen part of the deck, so the chances for unseen cards shift. The same principle is behind drawing specific cards from a known deck, where each card seen changes the counts for the rest. Without dependence, observation would teach you nothing. With it, the correct update follows from the counts, and the bookkeeping behind it is Bayes theorem.
A cautious point on games of chance: independence is the reason repeated plays of a casino game are not “due” to turn around, and why a house edge keeps acting on each bet. A shuffled deck being dealt out is a different, dependent process, but even then knowing about dependence does not turn a game with a built-in disadvantage into a reliable source of money. Outcomes remain uncertain, and gambling can cause harm. Anyone struggling should look to a local support service.
A quick test
To decide whether two events are independent, try these questions:
- If the first event happens, do the available outcomes for the second change? (Cards left in a deck, balls left in a bag.) If yes, they are dependent.
- Does a physical or mechanical process reset between the two events? (A die is rolled afresh; a deck is reshuffled.) If yes, they are likely independent.
- Compute P(B) and P(B | A). If they differ, the events are dependent.
If you are unsure, count the cases. For two dependent draws the number of possible ordered pairs is smaller than for independent ones: 52 × 51 = 2,652 versus 52 × 52 = 2,704, which is exactly where the denominators in the card example came from.