Independent events (definition and examples)
Independent events do not affect each other, so past results say nothing about the next one. Roulette and Coin Flip are independent; Mines tiles are not.
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Two events are independent when the outcome of one has no effect on the probability of the other. A fair coin does not remember its last toss, so each flip is a fresh 50-50. Most casino games are designed this way: every round starts from the same odds.
Independent: roulette and Coin Flip
Cashout Arcade's Roulette has 37 pockets, 18 of them red. After any history of spins, the chance of red on the next spin is 18 out of 37, or 48.65 percent. If the last nine spins were red, the chance of a tenth red is still 48.65 percent. The chance of ten reds in a row from a standing start is much smaller, (18/37) to the power 10, about 0.07 percent, but once nine have happened, only the last spin is left to be decided.
The same holds in Coin Flip, where you climb a ladder by calling heads or tails. Each flip is 50 percent. After five correct calls the multiplier is 31.04x, but the sixth flip is still a coin toss, not a harder or easier one.
Not independent: Mines
Some games change the odds as you go. In Mines, tiles are drawn without replacement from a fixed board of 25. With 3 mines, the first tile is safe with probability 22/25. If it was safe, the next tile is safe with probability 21/24, a little lower because one safe tile has been taken out. The events are dependent: what happened changes what remains.
| Situation | Independent? | Why |
|---|---|---|
| Roulette spin after any history | Yes | The wheel is reset each spin |
| Coin Flip call after a streak | Yes | Each toss is separate |
| Next safe tile in Mines | No | The board has fewer safe tiles left |
| Blackjack card in Cashout Arcade | Yes | Endless deck, nothing is used up |
| Blackjack card in a real single deck | No | Dealt cards leave the deck |
Cashout Arcade's Blackjack deals from an endless deck, so card counting does nothing there, which is not true at a physical table.
Common misconceptions
- "Independent means no streaks." Streaks happen all the time. Independence says only that a streak does not change the next chance. See Gambler's fallacy (definition and example).
- "Independent means the odds are even." They can be anything, such as 1 in 37. They just do not depend on history.
- "Two independent events: add the chances." To get the chance of both, multiply them.
For more on streaks see Gambler's Fallacy and Streak Myths in Casino Games. Coins are virtual and have no cash value.