# Equal and unequal probabilities

## In a nutshell

All events have a certain probability of occurring. If the probabilities for different events are equal, they are equally likely to happen. If not, then they are not equally likely to occur.

## Total probability

The probabilities of all outcomes of a trial must sum to $1$. If not, then this means there is another outcome with probability greater than $0$, and so this too is a possible event that might occur.

## Equal probabilities

Events with the same probability have an equal chance of occurring. When flipping a fair coin, the chance of it landing heads is equal to the chance of it landing tails, and so they have equal probability. As these are the only two possible events that occur, they each have a probability of $\dfrac{1}{2}$. Similarly, when rolling a fair six-sided die, the chance of it landing on any number is equal and so the probability it shows any given number is $\dfrac{1}{6}$.

## Unequal probabilities

The probabilities of events are not always equal. A fair die is more likely to show an even number than it is to show the number $5$, as there are more even numbers than there are $5$s. Although the only outcomes of rolling the die are either rolling an even number, or a $1$, $3$, or $5$, this does not mean that the probability of obtaining a $5$ is $\dfrac{1}{4}$. The chances of rolling a $1$, $3$ or $5$ are still the same and so these events have the same probability. However, this probability is less than the probability of rolling an even number.

##### Example

*Henry rolls a strange die, which has the number $3$ on $3$ sides, $2$ on $2$ sides, and $1$ on $1$ side. Are there any outcomes which have an equal chance of occurring?*

*The probability of rolling a $1$ is $\dfrac{1}{6}$.*

*The probability of rolling a $2$ is $\dfrac{2}{6} = \dfrac{1}{3}$.*

*The probability of rolling a $3$ is $\dfrac{3}{6} = \dfrac{1}{2}$.*

*Hence there are **no outcomes** which have an equal chance of occurring.*