The core idea
Two events are independent when the result of one has no influence on the chance of the other. A coin does not remember its last flip. A die does not remember its last roll. Each trial starts fresh.
A note on context: slots are games of chance with a built-in house edge, and no strategy or timing changes the odds. Most players lose money over time, and nobody should gamble money they cannot afford to lose. This article uses coins and dice because they are easy to picture, and the same reasoning explains why slot-style games cannot become “due.”
Coins: one flip at a time
For a fair coin, the probability of heads is 1/2 on every flip. That stays true however many heads or tails came before.
Suppose a fair coin has landed heads four times in a row. What is the chance the fifth flip is heads? Still 1/2. The coin has no way to know about the earlier flips, and no force pushes it back toward tails.
It can feel wrong, because a run of five heads seems unusual. The key is to separate two different questions:
- Before any flips, what is the chance of five heads in a row? That is 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32, which is 0.03125 or about 3.1%.
- Given that four heads have already happened, what is the chance the next flip is heads? That is 1/2.
The first question covers the whole sequence. The second asks only about one more flip. The unlikely part of “five heads” was already used up in the first four flips.
The multiplication rule
For independent events, the probability that both happen is found by multiplying:
P(A and B) = P(A) × P(B)
Examples with a fair die:
- Rolling a 6 twice in a row: 1/6 × 1/6 = 1/36, about 2.8%.
- Rolling an even number and then a number greater than 4: 3/6 × 2/6 = 6/36 = 1/6.
With two dice rolled together, each of the 36 combinations (such as a 3 on the first die and a 5 on the second) is equally likely, and the count of 36 comes directly from 6 × 6. The multiplication rule extends to longer chains, a topic discussed further under compound probability.
At least once
A useful trick for “at least one” questions is to compute the chance of none and subtract from 1.
What is the chance of rolling at least one 6 in four rolls of a fair die?
- Chance of no 6 on a single roll: 5/6.
- Chance of no 6 in four independent rolls: (5/6)^4 = 625/1296, about 0.482.
- Chance of at least one 6: 1 - 625/1296 = 671/1296, about 0.518.
So it is slightly better than even. Notice that it is not 4 × 1/6 = 2/3, which is what simply adding the chances would suggest. Adding works only for events that cannot overlap, and here a 6 can occur on several rolls.
Telling independent from dependent
Events are independent if knowing one happened does not change the other’s probability. Common cases:
| Situation | Independent? | Why |
|---|---|---|
| Two flips of a fair coin | Yes | Each flip is a separate physical event |
| Two dice rolled together | Yes | The dice do not interact |
| Two cards dealt from one deck | No | The first card changes what remains |
| Rain today and rain tomorrow | Usually not | Weather tends to persist |
The card case is explained in the article on conditional probability: when the chance of the second event depends on the first, the multiplication rule needs adjusting.
Independence and slot-style games
Modern slot games use random number generators to decide each spin. Every spin is generated separately, and the result of one spin does not alter the chance of the next. This is what independence means for a machine.
Several common beliefs fail because of it:
- A machine does not become “due” after a long dry spell.
- A machine that has just paid out is not “cooling down.”
- Changing seats, waiting or timing your play does not change the chances on the next spin.
These ideas are among the slot machine myths that keep circulating. They feel natural because people expect randomness to balance out quickly, which it does not, as discussed under the law of large numbers.
It is worth separating what is independent from what is designed. The outcome probabilities of a game are set by its designers and can be uneven; a symbol can be far rarer than another. But the chance of each outcome on a given spin is the same regardless of earlier spins.
Checking your intuition at home
You can test independence yourself with a coin:
- Flip a coin 100 times and write down the results.
- Look only at flips that come right after a head. Count how many of those are heads.
- Compare with flips that come right after a tail.
You should find roughly half in each case, though with only 100 flips the numbers will wobble noticeably. That wobble is a sign of how small samples behave, and a reminder that a few observations are a weak basis for conclusions. Running the experiment in a spreadsheet makes it easy to repeat many times.
Takeaways
- Independent events do not influence each other, so past results do not change future chances.
- Multiply probabilities for independent events that must all happen.
- For “at least one,” compute the chance of none and subtract from 1.
- Spins driven by a random number generator are modeled as independent, which is why no pattern of past spins can tell you what comes next.
Treat any gambling as paid entertainment, set a limit before you start, and never try to win back what you have lost. If it stops being fun, a national problem-gambling helpline or local health service can help.