# Calculating probability

## In a nutshell

Probabilities are used to indicate the chance that an event will occur. You can estimate the probabilities of events from experiments and the probability of any event will lie between $0$ and $1$.

## Terms

**Chance** | Something that cannot be foreseen. |

**Event** | A subset of outcome(s) of an experiment. |

**Random experiment** | Tests or trials, the results of which are random. |

**Mutually exclusive** | A property of multiple events, where both cannot occur at the same time. |

**Probability** | Indicates the chance that a specific result or event will occur. |

## Typical random experiments

**Probability of an event** | The proportion (relative frequency) of times an event is expected to occur when an experiment is repeated a large number of times. |

All probabilities range from $0$ (impossible) to $1$ (certain). All other probabilities lie on a scale in between. For example, if you repeatedly toss a fair coin, $\dfrac{1}{2}$ of the flips would land on heads, and so the probability of landing on heads is $\dfrac{1}{2}$. To analyse the probability of an event from an experiment, use the formula:

$P(A) = \dfrac{\text{number \ of \ times \ A \ occurs}}{\text{total \ number\ of \ trials}}$

##### Example 1

*Gina finds a coin on the floor but isn't sure if it is biased or not. In order to make an educated guess about its fairness, she flips the coin $50$ times and finds it lands on heads $38$ of these times. What is the most likely estimate of the probability that the coin lands on heads?*

$P(heads) = \dfrac{\text{number \ of \ heads \ thrown}}{\text{total \ number \ of \ throws}} = \dfrac{38}{50} = \underline{\dfrac{19}{25}}$

## Complementary probabilities

For any result from an experiment, it either happens or doesn't. The probability that an event ($A$) does **not **occur can be denoted as $P(A')$, $P(\overline{A})$ or $P(A^c)$ and is often referred to as the **complement** of $A$. The probability of an event happening or not happening always adds to total $1$. As these results are mutually exclusive, you have $P(A) + P(A')=1$. Rearranging this gives the formula that for any event $A$:

$P(A') = 1 - P(A)$

##### Example 2

*Andres rolls a fair die. What is the probability that it doesn't land on a $6$?*

*The probability that the die does land on a $6$ is $\dfrac{1}{6}$, as the die is fair.*

*$P(6) = \dfrac{1}{6}$*

*Use the formula given above:*

*$\begin{aligned}P(6')&= 1 - P(6) \\P(6') &= 1 - \dfrac{1}{6} \\P(6') &= \dfrac{5}{6} \end{aligned}$*

*Therefore, the probability that the fair die does not land on a $6$ is $\underline{\dfrac{5}{6}}$.*