Probability

Lesson

The sample space, sometimes called an *event space*, is a listing of all the possible outcomes that could arise from an experiment.

For example

- tossing a coin would have a sample space of {Head, Tail}, or {H,T}
- rolling a dice would have a sample space of {1,2,3,4,5,6}
- watching the weather could have a sample space of {sunny, cloudy, rainy} or {hot, cold}
- asking questions in a survey of favourite seasons could have a sample space of {Summer, Autumn, Winter, Spring}

Did you also notice how I listed the sample space? Using curly brackets { }.

An event is the word used to describe a single result from within the sample space. It helps us to identify which of the sample space outcomes we might be interested in.

For example, these are all events.

- Getting a tail when a coin is tossed.
- Rolling more than 3 when a dice is rolled
- Getting an ACE when a card is pulled from a deck

We use the notation, P(event) to describe the probability of particular events.

Adding up how many times an event occurred during an experiment gives us the **frequency **of that event.

The **relative frequency** is how often the event occurs compared to all possible events and is also known as the **probability of that event occurring**.

The probability values that events can take on range between 0 (impossible) and 1 (certain).

Calculating probabilities

We can calculate probabilities by constructing a fraction like this: $\frac{\text{what you want}}{\text{total }}$what you wanttotal which we write more formally as

P(event) = $\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}$total favourable outcomes total possible outcomes

An expected outcomes, as the name suggests, is the value (probability value) we expect from a given sample space or a set of given individual event probabilities.

The best way to describe this might be through an example.

A spinner is divided equally into $8$8 sections, but $3$3 of them are coloured green.

**a)** What is the probability of landing on green?

$\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}=\frac{3}{8}$total favourable outcomes total possible outcomes =38

**b)** If the spinner is spun $145$145 times, how many times would you expect it to land on green?

We take the probability of the event P(green) and multiply it by the number of trials.

P(green) x $145$145 = $\frac{3}{8}\times145=54.375$38×145=54.375

At this point we need to round appropriately, so we could say that if the spinner is spun $145$145 times we could expect it to be green $54$54 times.

A car manufacturer found that $1$1 in every $4$4 cars they were producing had faulty brake systems. If they had already sold $5060$5060 cars, how many of those already sold would need to be recalled and repaired?

Data collected in a certain town suggests that the probabilities of there being $0$0, $1$1, $2$2, $3$3, $4$4 or $5$5 or more car thefts in one day are as given in the table below.

What is the expected number of car thefts occurring on any particular day (to two decimal places)?

Car Thefts | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 or more |

Probability | $0.09$0.09 | $0.24$0.24 | $0.20$0.20 | $0.16$0.16 | $0.18$0.18 | $0.13$0.13 |

$260$260 fair dice are rolled.

What is the probability of getting an even number on a single roll of a die?

How many times would you expect an even number to come up on the $260$260 dice?