# Basics of probability

## In a nutshell

Probability indicates how likely an event is to occur. Probabilities can be described as 'certain', 'impossible' or 'likely', however it is better to give probabilities numerical values, so they can be compared or used in further calculations.

## Definition

Probability indicates the **chance** that an event will occur. It can be used to try and predict the outcome of random events.

All probabilities lie on a probability scale **between zero and one**, and can be expressed as a **fraction**, **decimal** or **percentage**. A probability of zero means an event will never happen whereas a probability of one means it will definitely happen. All events will have a probability between zero and one.

## Formula

Probabilities can be expressed as a fraction, decimal or a percentage.

The probability ($P$) of an event ($A$) occurring is given as:

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

##### Example 1

*What is the probability of rolling a $6$ on a fair die?*

*There is an equal probability of landing on each of the six sides of a die, hence:*

$P(6) = \underline{\dfrac{1}{6} = 0.1\dot{6} = 16.\dot{6} \%}$

Probabilities will also **always add up to one** since an event must either happen or not.

$P\text{(event\ happening)}+P\text{(event\ not\ happening)}=1$

##### Example 2

*A spinner has *$3$* different sized sides, one yellow, one red and one blue. What is the probability of the spinner landing on yellow? What is the probability of the spinner *__not__ landing on yellow?

#### Colour | **Yellow** | **Red** | **Blue** |

#### Probability | $x$ | $0.3$ | $0.1$ |

*First, work out the probability of the spinner landing on yellow:*

*$x+0.3+0.1=1\\ \ \\x=0.4$*

*So, the probability of the spinner landing on yellow is $\underline{0.4}$ or $\underline{40\%}$.*

*Then, work out the probability of the spinner **not** landing on yellow: *

*$P\text{(event\ happening)}+P\text{(event\ not\ happening)}=1\\ \ \\0.4+P\text{(event\ not\ happening)}=1\\ \ \\P\text{(event\ not\ happening)}=0.6$*

*So, the probability of the spinner **not** landing on yellow is *$\underline{0.6}$* or *$\underline{60\%}$*.*

## Probabilities of typical random events

Simple probabilities can be calculated for events such as tossing a coin, rolling a die or picking ball from a bag without looking.

##### Examples

**1. Tossing a coin and landing on heads: **

*There are two possibilities, so the probability is* $\dfrac{1}{2} = 0.5= 50 \%$

**2.** **Rolling a die and landing on** $1,2,3,4,5\ or\ 6$**: **

*There are six possibilities, so the probability is*** **$\dfrac{1}{6} = 0.1\dot{6} = 16.\dot{6} \%$

**3. Spinning a **$5$**-sided spinner and landing on side A, B, C, D or E:**

*There are five possibilities, so the probability is*** **$\dfrac{1}{5} = 0.2 = 20 \%$

**4. Picking a red marble from a marble bag with **$30$** marbles, **$10$** of each colour (red, green and blue):**

*There are three possibilities, so the probability is* $\dfrac{1}{3} = 0.3\dot{3} = 33.\dot{3} \%$