Some probability mistakes are about psychology, such as expecting a result to be “due”. Others are plain arithmetic: the right question, the wrong calculation. This article covers three calculation errors that appear again and again, with small examples you can check by hand. All numbers are illustrations of the method, not advice about any particular game.
Mistake one: adding when you should multiply (or the reverse)
The most common slip is adding probabilities in a situation where they do not simply add.
Suppose you roll a fair die twice and want to know the chance of at least one six. A tempting answer is 1/6 + 1/6 = 1/3. The pattern is easy to see as wrong if you extend it: with six rolls it would give 6/6, a certainty, yet six rolls clearly cannot guarantee a six.
The reliable route is to work out the chance of the opposite outcome and subtract from one. This is the complement rule. The chance of no six on one roll is 5/6. For two separate rolls the chances multiply: 5/6 × 5/6 = 25/36. So the chance of seeing a six at some point is 1 − 25/36 = 11/36, which is about 30.6 per cent. That is a little under one third, not equal to it.
A rule of thumb helps decide which operation to use:
- Use multiplication when you need one thing and another to both happen, in independent steps.
- Use addition when you want one outcome or another, and the two outcomes cannot happen together.
- If the outcomes can overlap, plain addition double counts, which brings us to the next mistake.
Mistake two: double counting overlaps
Take one card from a standard 52-card deck. What is the chance it is a heart or a king?
Adding the two groups gives 13 hearts + 4 kings = 17 cards, so 17/52. But the king of hearts sits in both groups and was counted twice. Subtracting it once gives 13 + 4 − 1 = 16 cards, so 16/52, which simplifies to 4/13.
The general idea is called the inclusion-exclusion principle: add the sizes of the groups, then subtract what they share. It matters most in games where categories overlap, such as “a pair or a flush” or “a critical hit or a status effect”.
A good habit is to ask, “Can both of these happen at once?” If yes, check for overlap before adding.
Mistake three: treating dependent events as independent
Multiplying probabilities is only correct when the first event does not change the second. With cards drawn without replacement, it does.
Illustration: draw two cards from a standard deck and ask for the chance both are aces.
- Wrong method (pretending the draws are independent): 4/52 × 4/52 = 1/13 × 1/13 = 1/169.
- Correct method: the first card is an ace with probability 4/52. If it was, 3 aces remain among 51 cards, so the second is an ace with probability 3/51. Multiplying: 4/52 × 3/51 = 12/2652 = 1/221.
The correct answer is smaller than the wrong one, because removing an ace from the deck makes a second ace less likely. This is the idea behind dependent events, and it is why conditional probability, the chance of one thing given that another has happened, is the working tool for card problems. The same thinking applies to any game that draws from a shrinking pool, including loot tables with no replacement.
The reverse error also happens. People sometimes assume that independent events are linked, for instance believing a roulette spin is influenced by the previous one. Dice, coins and roulette spins are separate trials. Checking which situation you are in is the first step of any calculation.
Two smaller slips worth knowing
Mixing up percentages and odds. “Odds of 1 to 3” and “a 1 in 3 chance” are different statements: the first corresponds to a probability of 1/4. Always convert to a probability before combining numbers, and keep odds and percentages straight when reading anyone else’s figures.
Forgetting the order of questions. “The chance of two sixes in two rolls” and “the chance of a six somewhere in two rolls” ask different things, and answering the wrong one gives a confident, wrong answer. Write the question out in words first.
A checking routine
Before trusting a probability you have worked out, run through these steps:
- Does the answer lie between 0 and 1 (or 0 and 100 per cent)?
- Do the pieces add up? The probabilities of all possible outcomes should sum to exactly 1.
- Does the answer make sense compared with a rough estimate?
- If you added, could the outcomes overlap? If you multiplied, are the events truly independent?
None of this makes any game favourable. Correct arithmetic simply shows what the odds actually are, and for casino games, that picture includes a house edge and uncertain outcomes. Understanding the numbers properly is a protection against misleading claims, not a route to winning.