Probability

UK Secondary (7-11)

Theoretical Probability

Lesson

The sample space, sometimes called and *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).

Calculatingprobabilities

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

The $26$26 letters of the alphabet are written on pieces of paper and placed in a bag. If one letter is to be picked out of the bag at random what is the probability of picking a:

J?

K, Y or R?

Letter in the word PROBABILITY?

M, K, D, O, H or B?

Letter in the word WORKBOOK?

A cube has six faces. Each face is painted a certain colour and the cube is rolled. How many faces should be painted blue so that the probability of blue appearing on the uppermost face is:

$\frac{1}{2}$12?

$\frac{1}{3}$13?

$\frac{1}{6}$16?

$1$1?

A book has pages numbered from $1$1 to $100$100. If the book is opened to a random page, what is the probability that the page number:

Is a multiple of $9$9?

Has digit ‘$6$6’ in the page number?