Probability Basics and Simple Real-World Data

Section 1What probability means

Probability is a number that describes how likely an event is. It can be written as a fraction, decimal, or percentage. A probability of 0 means impossible, a probability of 1 means certain, and values between 0 and 1 show different levels of chance.

Section 2Outcomes and events

An outcome is one possible result of a trial, such as rolling a 4 on a die. An event is a set of outcomes, such as rolling an even number. Easy to mix up - the event can include one outcome or several outcomes.

Section 3Sample spaces

A sample space lists all possible outcomes of a trial. For one fair die, the sample space is {1, 2, 3, 4, 5, 6}. Listing the sample space first helps you avoid missing outcomes or counting impossible ones.

Section 4Equally likely outcomes

When outcomes are equally likely, use probability = favourable outcomes / total outcomes. For example, the probability of rolling an even number on a fair die is 3 / 6 = 1 / 2. Watch out: this rule depends on the outcomes being equally likely.

Section 5Complements

The complement of an event is the event not happening. Since probabilities in a complete sample space add to 1, P(not A) = 1 - P(A). Here's the trick: complements are often faster than listing many unwanted outcomes.

Section 6Relative frequency

Relative frequency uses real results to estimate probability. It is calculated as number of times the event happens divided by total trials. For example, if rain falls on 12 out of 30 recorded school days, the relative frequency of rain is 12 / 30 = 0.4.

Section 7Interpreting real-world data

Real-world data are not always perfectly balanced like dice or coins. A small sample can be misleading, and bias can affect results. In plain words, probability from data is an estimate, so always ask how the data were collected and whether the sample is large enough.