Probability

When an experiment has more than one step, the sample space can be represented in several ways:

• by listing all possible ordered outcomes;
• using a table (best for two-step experiments);
• using a tree diagram (useful for any number of steps).

In a random experiment, the probabilities of all the outcomes add up to \(1\): $$\textcolor{colorprop}{P(\text{outcome}_1) + P(\text{outcome}_2) + \dots + P(\text{outcome}_n) = 1}$$

The probability of an event is the sum of the probabilities of its outcomes: $$\textcolor{colorprop}{P(\text{event}) = \text{sum of the probabilities of its outcomes}}$$

For any event \(E\) and its complementary event \(E'\), their probabilities must add up to \(1\) (or \(100\pourcent\)): $$\textcolor{colorprop}{P(E) + P(E') = 1}$$ This leads to the useful formula for finding the probability of the complement: $$\textcolor{colorprop}{P(E') = 1 - P(E)}$$

1. Determine the event: describe it, then list its outcomes — write it as a set.
2. Compute its probability:
   • if the outcomes are equally likely: \(P(A) = \dfrac{\text{number of outcomes of } A}{\text{total number of outcomes}}\);
   • otherwise: add the probabilities of the outcomes of the event;
   • when the complementary event is simpler: \(P(A) = 1 - P(A')\).

A Venn diagram gives the elements of each region, or their number written in brackets: \((9)\) is a count, a bare \(9\) is an element. The rectangle \(U\) holds the whole group of \(N\) elements. When one element of \(U\) is chosen at random, the \(N\) elements are equally likely, so $$P(\text{event}) = \frac{\text{number of elements in the regions of the event}}{N}.$$

• \(A \cap B\) (“\(A\) and \(B\)”): the overlap of the two circles;
• \(A \cup B\) (“\(A\) or \(B\)”, at least one of the two): every region inside the circles, each counted once;
• “\(A\) only”: the part of the circle \(A\) outside \(B\); “exactly one of the two”: \(A\) only and \(B\) only together;
• \(A'\) (“not \(A\)”): every region outside the circle \(A\); “neither \(A\) nor \(B\)”: the region outside both circles.

Check that the numbers of the four regions add up to \(N\).

A two-way table counts the \(N\) individuals of a group by two characteristics. When one individual is chosen at random from the group, the \(N\) individuals are equally likely, so the probability of an event is the number of individuals it counts divided by \(N\), the total in the bottom-right cell:

• for one category, use the total of its row or of its column;
• for “\(A\) and \(B\)”, use the cell where the row of \(A\) meets the column of \(B\);
• for “\(A\) or \(B\)” (at least one of the two), add the two totals and subtract the shared cell, whose individuals would otherwise be counted twice;
• for “not \(A\)”, use the complement rule, or the other totals.

The same fractions have a second use. When the table records a representative sample of a larger population, each fraction is a frequency of the sample, which estimates the probability for a person chosen from the whole population. When the table records the results of repeated trials of an experiment, \(N\) counts trials, not individuals: each fraction is an observed frequency, an experimental probability, which estimates the probability for another trial of the same experiment.

An experiment is repeated \(n\) times under the same conditions, each repetition having no influence on the others, and the frequency of an event of probability \(p\) is observed.

• Fluctuation: for a fixed \(n\), the frequency can change from one series to another. Two series of the same length do not necessarily give the same frequency.
• Stabilisation: when \(n\) grows, the frequencies tend to move closer to \(p\). Long series usually stray less from \(p\) than short ones.
• Expected count: over \(n\) repetitions, the expected count of the event is \(n \times p\); the observed count is about \(n \times p\), in the sense that the observed frequency, the count divided by \(n\), is usually close to \(p\). “About”: the observed count is not known in advance, and it need not equal \(n \times p\) exactly.