How do I calculate a simple probability?
Divide the number of favourable outcomes by the total number of equally likely outcomes. Drawing a heart from a standard deck is 13 divided by 52, or 0.25 — a 25% chance. The phrase equally likely is doing real work: the formula only holds when every outcome has the same chance, which is why it applies to fair dice and shuffled cards but not to weather or sporting results.
What is the difference between independent and dependent events?
Independent events do not affect one another — two coin flips, or two dice. Multiply their probabilities to get the chance of both. Dependent events do affect one another, typically because you are drawing without replacement: the chance of a second heart is 12 out of 51, not 13 out of 52. Treating dependent events as independent is the most common error in probability, and it always produces an answer that is too clean.
How do AND and OR differ?
AND means both must happen: multiply. OR means at least one happens: add the individual probabilities, then subtract the overlap, because otherwise you count the both case twice. For two mutually exclusive events there is no overlap so simple addition works. Forgetting to subtract the overlap is why naive OR calculations sometimes exceed 100%, which is a useful signal that something has gone wrong.
Why is at least one easier to calculate backwards?
Because the opposite of at least one is none, and none is a single straightforward multiplication. To find the chance of at least one six in four rolls, calculate the chance of no six — five-sixths to the fourth power, about 0.482 — and subtract from one, giving roughly 0.518. Computing it directly means adding the cases for exactly one, two, three and four sixes, which is far more work for the same answer.
What are odds, and how do they relate to probability?
Odds compare favourable to unfavourable outcomes; probability compares favourable to total. A one-in-four probability is odds of 1 to 3, not 1 to 4. Betting markets quote odds, statistics quotes probability, and conflating them shifts the number meaningfully. To convert, probability equals odds-for divided by the sum of both sides.
Does probability tell me what will happen?
No, and this is where intuition misleads most reliably. A 70% chance means that across many identical trials the event happens about 70% of the time; it says nothing about the next single trial. Independent events also have no memory — a coin that has landed heads five times is still 50/50, and believing otherwise is the gambler's fallacy. Small samples routinely look nothing like the underlying probability.