Probability Basics

A probability is simply a number between 0 and 1 that says how likely something is: 0 means it can't happen, 1 means it's certain, and 0.5 means a coin flip. The first formula is just counting: the outcomes you care about (the ones in $A$) over all the possible outcomes. The second, $P(A') = 1 - P(A)$, says the chance something doesn't happen is whatever's left over. That's really all there is to start with; the rest of the unit builds on these two.

Vocabulary:

• Experiment: a process with uncertain results, like rolling a die.
• Sample space $S$: all possible outcomes, $\{1, 2, 3, 4, 5, 6\}$.
• Event: a set of outcomes, like "roll an even number" $= \{2, 4, 6\}$.

The law of large numbers. As an experiment is repeated, the relative frequency of an event settles toward its probability. A fair coin won't give exactly half heads in 10 flips, but it will in the long run. This is what lets a probability mean something real: "a 3% chance" turns into "about 60 out of 2,000." There's an animation further down that shows the fraction of heads settling in.

The complement rule is often the shortcut: "at least one" is $1 - P(\text{none})$. Think about why. "At least one head in 5 flips" can happen in lots of ways (1 head, 2 heads, ...), but its opposite, "no heads," happens in exactly one way. It's much easier to find the chance of the one way and subtract.

A common mistake. After 7 heads in 10 flips, it's tempting to think tails is now due, so the coin can "even out." It isn't. The coin has no memory, and the next flip is still 50-50. The fraction of heads drifts toward 1/2 because the early streak gets swamped by hundreds of later flips, not because tails start catching up.

Insurance premiums, weather forecasts ("70% chance of rain") and casino house edges are long-run probabilities.