What independent events actually mean
Independent events do not remember the past. See what that means for coins, dice and cards, and what it does not mean.
A coin does not owe you tails because the last three flips were heads. A die does not remember the last face. When knowing one result does not change the probabilities of the next, the trials are independent. The definition is small. People still talk past it.
This article is for adults 18 and older. It is about separate trials, not a staking method. It does not say any game is worth your money. Laws on wagering differ by place.
A clean definition
Two events are independent when knowing one does not change the probability of the other. If a fair coin shows heads, the next flip is still one half heads and one half tails, assuming the coin and the toss stay fair. If a fair six-sided die shows a 4, the next roll is still one chance in six for each face.
Independence is a property of the setup, not a mood. It fails when the setup changes. Cards dealt from a shrinking deck are the classroom example. Four aces sit in a 52-card deck, so the first ace has probability 4/52, which is 1/13. After that ace is removed, three aces remain in 51 cards, so the next-card chance is 3/51, which is 1/17. The deck has a memory because the card is gone. A coin does not lose a face when it lands.
Multiplying chances
When events are independent, multiply their probabilities to get the chance they all happen. Two heads in a row on a fair coin: 1/2 times 1/2 equals 1/4. Three specified heads in a row: 1/8. Five heads in one specified row of five tosses: 1/32.
Those fractions describe that specific sequence, not "a streak somewhere in a long afternoon." In a long series you should expect some runs. Their appearance does not rewrite the next toss. Expecting the opposite result to balance the recent past is the error unpacked in the gambler's fallacy.
Check the arithmetic before you trust a story. Three independent events at 1/2 are not "about even." They are 1/8 together. Four of them are 1/16. The drop is multiplication, and multiplication gets small quickly.
Games that fit, and games that do not
A fair roulette spin is built so the last color does not change the next one. Zeros change the price of the bets. They do not give the wheel a memory. A blackjack shoe is different in a limited way: dealt cards stay out until the shuffle, so the next card is not fully independent. That effect is real and usually small beside the stories told about a "hot" table. Hot and cold describe the past. They are not a force on the next hand.
Sports results are a weaker fit for the coin model. Teams, rest and injuries carry from one day to the next, so games are not fresh flips of one coin. You can still reject a fake version of the idea. One upset does not entitle anyone to a corrective win the following week. Extreme stretches cool off without magic, which is the point of regression to the mean.
A long average and a short night
Independence helps explain a result that feels contradictory. The long-run average can sit near a known value while a single night swings hard. Each trial adds its own luck. Over many trials the average settles near the expected value. Over twenty trials it often does not. Expected value is that long-run average for one bet, not a forecast for the next hour.
If a price is worse than the fair chance, repeating independent trials does not grind the gap away. It makes the gap easier to see. Volume is not a cure for a bad price.
Use the idea as a filter. If a story says the table is due, ask whether the mechanism actually remembers. If it does not, the story is color, not information. Keep any stake inside money you can lose, and stop when the limit you set in advance arrives. If play is hurting sleep, relationships or bills, stop and look for local help.
Frequently asked questions
Are shuffled cards independent? Draws without replacement are not fully independent. Removing a card changes what remains. After a thorough shuffle restores the full deck, the previous shoe no longer affects the next card.
Does a fair coin balance itself? No. A fair coin does not store the last result. Five heads in a row, on those five tosses, had probability 1/32 before they happened. The next toss is still one half.
If trials are independent, is the bet fair? No. Independence says trials do not influence each other. The posted price can still be worse than the true chance. A house margin and independence often sit in the same game.