SL 4.5—Probability basics

A trial produces an outcome; the sample space $U$ lists all possible outcomes, and an event $A$ is a subset of $U$. For equally likely outcomes, $P(A)=n(A)/n(U)$ and $P(A')=1-P(A)$.

Represent $U$ with a complete list or outcome table. Relative frequency estimates probability from repeated trials and may differ from the theoretical value; with more trials it can become more stable but is not guaranteed to equal it.

Example

For two fair coins, $U=\{HH,HT,TH,TT\}$. The event of exactly one head is $\{HT,TH\}$, so its probability is $2/4=1/2$. If this probability applies to 128 trials, the expected number is $128(1/2)=64$.

The formula $n(A)/n(U)$ requires equally likely outcomes. An expected count such as 12.8 is a long-run average, not a claim that 0.8 of a person will occur in one trial.