Probability as a number
The likelihood of an event after a trial can be placed on a spectrum from 0 to 1 using fractions or decimals, or from 0\% to 100\% using percentages:
A probability can never be less than 0 or more than 1. The larger the number, the more likely it is, and the smaller the number, the less likely it is. We will now look at how to determine these numbers exactly.
Previously we looked at the difference between an outcome and an event.
An outcome represents a possible result of a trial. When you roll a six-sided die, the outcomes are the numbers from 1 to 6.
An event is a grouping of outcomes. When you roll a six-sided die, events might include "rolling an even number", or "rolling more than 5".
Each outcome is always an event - for example, "rolling a 5" is an event.
But other events might not match the outcomes at all, such as "rolling more than 6".
If every outcome in a trial is equally likely, then the probability of one particular outcome is given by the equation:\text{Probability} = \dfrac{1}{\text{Size of sample space}}
For instance, the probability of rolling a 4 on a 6-sided die is \dfrac{1}{6}, since there are 6 possible numbers on a die which are equally likely, and only 1 of them is a 4.
Remember that the sample space is the list of all possible outcomes. We can multiply this number by 100\% to find the probability as a percentage.
Examples
Example 1
A probability of \dfrac{4}{5}\, means the event is:
An outcome represents a possible result of a trial.
An event is a grouping of outcomes.
If every outcome in a trial is equally likely, then the probability of one particular outcome is given by the equation:\text{Probability} = \dfrac{1}{\text{Size of sample space}}
Unequal probabilities
If the outcomes in a sample space are not equally likely, then we have to think about splitting the sample space up into "favourable outcomes" and the rest. Then we can use the formula:\text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}
If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.
Examples
Example 2
A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.
If the probability of picking a red marble is \dfrac{4}{10}, how many red marbles are there in the jar?
What is the probability of picking a blue marble?
\text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}
If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.
Complementary events
We can also use a useful fact about complementary events - since exactly one of them must happen, their probabilities always add to 1. This means if we know the probability of an event, the probability of the complementary event will be one minus the probability of the original: \text{Probability of complementary event} = 1 - \text{Probability of event}
Examples
Example 3
The probability of the local football team winning their grand final is 0.36.
What is the probability that they won't win the grand final?
If two events are complementary then their probabilities will add to 1. This means:\text{Probability of complementary event} = 1 - \text{Probability of event}