Probability vs Relative Frequency

Every probability is a number in the interval from 0 to 1, and the probabilities of all the outcomes in the sample space add to exactly 1. $P(A)$, read "the probability of A", is the long-run relative frequency of the event $A$: the share of trials on which $A$ happens, once the number of trials is large. When every outcome is equally likely you can reach that number by counting, dividing the outcomes in $A$ by the total, but counting is a special case rather than the definition.

Draw a single digit at random from 0 through 9, each with probability 0.1. Let $A$ be "the digit is 0, 1, or 2". Then $P(A) = 3/10 = 0.30$.

"A probability of 0.30 means three out of every ten." Read ten digits from a random digit table and the chance that exactly three of them fall in $A$ is 0.2668, so that reading is wrong about 73 percent of the time, and roughly 2.8 percent of ten-digit runs contain none from $A$ at all. A probability is a rate the results approach over a long run, not a quota each block of ten has to fill. What the long run does and does not promise is the content of the law of large numbers.

Two boundaries are worth naming. Counting outcomes gives the right answer only when the outcomes really are equally likely, which rules out a loaded die or a thumbtack. And probability 0 does not always mean impossible: for a continuous random variable every single exact value has probability 0, yet the variable still lands somewhere.

Probability is introduced in topic 2.4, Introduction to Probability, after simulation has already given you an empirical version of the same quantity.

A relative frequency turns a count into a share of the whole: divide the number of observations in a category by $n$, the total number of observations. The result always lands between 0 and 1, and multiplying by 100 reports the same quantity as a percent. When the categories are mutually exclusive and cover every observation, the relative frequencies add to exactly 1.

Forty students report how they get to school: 18 walk, 14 take the bus, 8 come by car. The relative frequencies are $18/40 = 0.45$, $14/40 = 0.35$, and $8/40 = 0.20$, and those add to 1.00. As percentages, 45 percent, 35 percent, and 20 percent.

The mistake relative frequency exists to prevent is comparing raw counts across groups of different sizes. "School B had 30 walkers to our 18, so walking is more common there" collapses if School B has 200 students: $30/200 = 0.15$ against $18/40 = 0.45$. The count is larger and the share is a third as big. A smaller slip is reporting the relative frequency as 45 when the value is 0.45. The proportion and the percent are the same quantity on two scales, so say which one you are using.

The shares only add to 1 when each observation is counted once. Let students name every method they use and someone who walks and buses appears twice, the shares total more than 1, and the denominator has stopped being the number of students. In a two-way table one word covers three different divisions, by the grand total, by a row total, or by a column total, so a relative frequency there means nothing until you state what you divided by.

Relative frequency is also how a probability gets estimated from data: run a random process many times and the share of trials producing an outcome settles near its probability as the number of trials grows. That share is an estimate built from one set of trials, not the probability itself. Summary statistics for one categorical variable are Unit 1 topic 1.3.