Mathematics from zero
Combining events
One coin flip has a clear probability of heads: a half. But flip two coins — what is the chance both land heads? You already know each one’s probability. The new skill is combining them.
After this lesson you can find the probability of two independent events both happening, of two exclusive events where either happens, and of an event not happening.
Two events are independent when one does not affect the other. Flipping a coin and then flipping it again — the first result changes nothing about the second. Such events are independent. The “and”, “or”, “not” of the logic unit return here, now applied to probabilities rather than to true and false.
For two independent events both happening, multiply their probabilities. The
chance of heads is 1/2. The chance of heads and then heads again is
1/2 × 1/2 = 1/4. Multiplying makes sense: only a quarter of all two-flip outcomes are
heads-heads, because each flip cuts the surviving possibilities in half.
For two events that cannot both happen, add their probabilities for “either”. A
single die cannot show both 1 and 2 on one roll — those outcomes are mutually
exclusive. The chance of rolling a 1 or a 2 is 1/6 + 1/6 = 2/6. When events
cannot overlap, “or” adds their separate probabilities.
The probability of an event NOT happening is 1 minus its probability. This is the
complement. If the chance of rain is 0.3, the chance of no rain is
1 − 0.3 = 0.7. It must work this way: the event happens or it does not, so the two
probabilities are the whole of certainty, and certainty is 1.
You flip a fair coin twice. What is the probability of getting heads both times?
Each flip is independent — the first result does not touch the second. The probability
of heads on one flip is 1/2.
You want heads on the first and heads on the second. For two independent events both
happening, multiply: 1/2 × 1/2.
1/2 × 1/2 = 1/4, which as a decimal is 0.25. So there is a 1-in-4 chance of two
heads. Check by listing the four equally likely two-flip outcomes — heads-heads,
heads-tails, tails-heads, tails-tails — and exactly one is heads-heads.
Why this works
Why does “and” multiply while “or” adds? Because they shrink and grow the possibilities in opposite ways. Requiring two things to both happen keeps only the overlap — a slice of a slice, which multiplication produces. Accepting either of two non-overlapping things gathers both slices together — which addition produces. The logic words AND and OR carry straight over into probability.
Common mistake
A common mistake is adding probabilities when events must both happen — saying two
heads is 1/2 + 1/2 = 1. That would mean two heads is certain, which is plainly wrong.
“Both happen” multiplies; only “either happens”, for exclusive events, adds. Check
which question you are answering before choosing the operation.
Two coin flips. Probability of heads both times is 0.5 × 0.5. Type it as a decimal.
Roll a die. Probability of a 1 or a 2 is 1/6 + 1/6. Type the numerator of the sum over 6.
The chance of heads is 0.5. What is the chance of NOT heads? Type it as a decimal.
Two dice. Probability both show 6 is 1/6 × 1/6. Type the denominator of the product.
The chance of rain is 0.3. What is the chance of no rain? Type it as a decimal.
Two independent events each have probability 1/2. What is the probability that both happen?
Probabilities combine with three moves. For two independent events both happening, multiply their probabilities. For two mutually exclusive events where either happens, add their probabilities. For the probability of an event not happening — its complement — subtract its probability from 1. The “and”, “or”, “not” of logic carry straight into probability.