Basic probability
Probability is a measure of chance. It tells us how likely an event is to happen. We can use words such as certain, likely, evens and impossible to describe the likelihood of events.
If an event is certain to occur, that means it must happen 100% of the time. An example of this would be rolling a number less than 9 on a six-sided dice
. This is certain as every number on a standard dice is less than 9.
More usually in mathematics, we denote probabilities in decimal or fraction form. This means that if something is 'certain' to happen, it would have a probability of 1.
Similarly an 'impossible' event can never occur, such as the chance of pulling a black ball from a bag of green balls. It simply cannot happen. In decimal notation, this is a probability of 0.
The terms 'likely' and 'unlikely' mean a probability of greater than and less than 0.5 respectively. We can show this information on a probability diagram:
Roll of a dice
In the example in the previous section, we talked about the probability of rolling a number less than 9 and decided it was certain.
What about the probability of getting a number 4 on a single roll of a dice? If we know that the dice is fair, the probability of rolling a 4 is \(\frac{1}{6}\). This is because we are looking for one specific number out of a total of 6 numbers.
The probability of rolling any single number on a normal dice is \(\frac{1}{6}\).
What if we had a 20-sided dice, and we wanted to know the probability of getting a number less than 5? There are 4 numbers less than 5 (1,2,3,4) and the dice has 20 numbers in total. The probability is therefore \(\frac{4}{20}\).
If we are looking for the probability of not rolling a number less than 5 (or in other words, rolling a number greater than or equal to 5) then we can use the following formula:
\(\text{Probability~of~an~event~not~happening}\) \({=}~{1}~â~\text{the~probability~of~it~happening}\)
P (not rolling a number less than 5) = 1 â \(\frac{4}{20}~=~\frac{16}{20}\).
Probability in other events
What about something less simple than rolling a dice? Let's look at the probability of Cardiff City FC scoring a goal in their next game. We can estimate the probability of this event happening by looking at the relative frequency.
\(\text{Relative~frequency}~=~\frac{number~of~successes}{number~of~trials}\)
This equation means that if we look at the last 10 games that Cardiff City played, and noticed that they scored in 6 of those games, then the relative frequency would be \(\frac{6}{10}\).
We can use relative frequency as an estimation of the probability of an event happening in the future. In our example, we would estimate the probability of Cardiff City scoring in their next game to be \(\frac{6}{10}\).