Probability and its properties: Laplace, complement and union

What probability measures

Imagine that you repeat a random experiment many times and count how often an event $S$ occurs. That quantity is the absolute frequency $f(S)$. If you repeat the experiment $N$ times, the relative frequency is

$$\mathrm{fr}(S)=\frac{f(S)}{N}$$

When $N$ is very large, the relative frequency settles down around a fixed value. That value is the probability $P(S)$. This behaviour is known as the law of large numbers. That is why, in practice, a relative frequency obtained from a lot of data serves as an estimate of the probability.

Relative frequencies have three obvious properties: they lie between 0 and 1, that of the certain event is 1, and that of two events that cannot occur together is the sum of the two. The axioms of probability are exactly those three ideas.

Laplace's rule

When all the elementary outcomes of the sample space are equally likely, calculating a probability is a matter of counting.

A coin, a die or the draw of a card from a well-shuffled pack are equally likely. A biased coin or a loaded die are not. Before using Laplace's rule, check that all the outcomes have the same probability. If the question says "at random", they do.

To count favourable and possible outcomes, use what you saw in the previous section: listing the outcomes, a Venn diagram or an outcome table for two experiments (for example, two dice). In an outcome table, each cell is an equally likely elementary outcome.

Properties of probability

Why they are true:

• Complement. $A$ and $A^{\prime }$ do not overlap and together cover the whole sample space, so $P(A)+P(A^{\prime })=P(U)=1$. It is the most useful property: when a question asks for "at least one", it is usually best to calculate the opposite.
• Union. If you add $P(A)+P(B)$, the common region $A\cap B$ is counted twice, so it must be subtracted once.
• Inclusion. If $A$ lies inside $B$, then $B$ contains $A$ and also other outcomes, which do not reduce the probability.

The four regions of a Venn diagram

Two events $A$ and $B$ divide the sample space into four regions that do not overlap. Their probabilities add up to 1, and from this almost all the relationships follow:

Three formulas that come from this drawing are used very often:

$$P(A)=P(A\cap B)+P(A\cap B^{\prime })$$

$$P(A\cap B^{\prime })=P(A)-P(A\cap B)$$

$$P(A^{\prime }\cap B^{\prime })=1-P(A\cup B)$$

The first says that $A$ splits into the part it shares with $B$ and the part it does not. The third says that "neither $A$ nor $B$" is the opposite of "$A$ or $B$". In the same way, $P(A^{\prime }\cup B^{\prime })=1-P(A\cap B)$.

Contingency tables

A contingency table (or two-way table) classifies the data according to two criteria at once and gives the row and column totals. Each interior cell is an intersection, and the marginal totals give the simple events. If you choose an element at random, Laplace's rule applies directly: the probability of an event is the number of cases of the event divided by the grand total.

For example, in this table of 50 customers of a shop:

The probability that a customer is under 30 and pays by card is $\frac{12}{50}$: the interior cell divided by the grand total. The probability that they pay in cash is $\frac{24}{50}$: the column total divided by the grand total. The probability of "under 30 or pays by card" is $\frac{20}{50}+\frac{26}{50}-\frac{12}{50}=\frac{34}{50}$.