4.5 Probability basicsIB Maths: Applications and Interpretation SL: Revision notes

A trial is one performance of an experiment, such as rolling a die once. Each possible result is an outcome. The set of all possible outcomes is the sample space, written $U$. An event is a set of outcomes we are interested in, for example 'an even number'. Outcomes are equally likely when none is favoured, as with a fair coin or a fair die. The sample space can be shown as a list or a table. For a die and a coin, $U$ has $6\times2=12$ outcomes: (1, H), (1, T), ..., (6, T).

For equally likely outcomes, the probability of an event $A$ is $P(A)=\frac{n(A)}{n(U)},$ where $n(A)$ is the number of outcomes in $A$ and $n(U)$ the number in the sample space. A probability lies between 0 (impossible) and 1 (certain). Example: a bag holds 5 red, 3 blue and 2 green counters. $P(\text{blue})=\frac{3}{10}$.

The complement of $A$, written $A'$ ('not $A$'), is every outcome not in $A$. Since $A$ and $A'$ together make up the whole sample space, $P(A')=1-P(A).$ This is the quickest route when 'not' is easier than counting directly: $P(\text{not green})=1-\frac{2}{10}=\frac45$.

Theoretical probability comes from counting equally likely outcomes, for example $\frac14$ for each sector of a fair four-sector spinner. Experimental probability is the relative frequency found from trials: $\text{relative frequency}=\frac{\text{number of times the event occurs}}{\text{number of trials}}.$ If a spinner lands on sector 2 in 62 of 200 spins, the relative frequency is $0.31$. As the number of trials grows, relative frequency usually gets closer to the theoretical probability, so a large gap after few trials is not strong evidence of bias. Simulations (coins, dice, cards or random numbers on a GDC) can be used to model real situations.

If an event has probability $p$ and there are $n$ trials, the expected number of occurrences is $n\times p.$ Example: 128 students, probability of absence $0.1$, so the expected number absent is $128\times0.1=12.8$. The expected value need not be a whole number: it is a long-run average, not a prediction of one outcome.