Probability

UK Secondary (7-11)

Describing Chance

Lesson

**Probability **is all around us.

- There are the weekly Lotto and Powerball draws.
- Poker machines in clubs.
- Many games that use dice, coins and cards.
- Weather
- Safety

The results of all of these are determined by **probability **- also called **chance**.

In day to day life there are also many other places where the language of probability is also used.

- The chance of rain today is 75%.
- A medication has an effectiveness of 90%
- 4 in 5 people agree that a particular brand of toothpaste is more effective than another.
- Most likely the supermarket would have the brand of milk I want to purchase.

A probability continuum is one way to visualise the scale of likelihoods.

**CERTAIN**is when something will definitely occur. We assign a value of 1, or 100% to the probability of certain. For example, if I were to roll a standard dice, it is certain that I will roll a number less than 10.**IMPOSSIBLE**is when something can not occur. We assign a value of 0, or 0% to the probability of impossible. For example, if I were draw a card from a standard pack of playing cards, it is impossible that I would pull out a card with a 27 on it!- Exactly half way along this continuum, we find the value of 1/2, or 50%. We call this
**EVEN CHANCE**, or**EQUALLY LIKELY**. For example, if I were toss a fair coin I could get a head or a tail. Both of these have an even chance of occuring. - Anywhere between impossible and even chance (values between 0 and 50%) we call not likely, or
**unlikely**. - Anywhere between even chance and certain (values between 50% and 100%) we call
**likely**.

Here is another visual explanation of the probability continuum.

An experiment or trial are the words used to describe the event or action of doing something and recording results. For example, the act of drawing cards from a deck, tossing a coin, rolling a dice, watching the weather, asking questions in a survey or counting cars in a carpark could all be examples of experiments or trials.

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 of an experiment. 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 the name given to the probability of that event happening.

Lets look at a situation and identify the *experiment*, *sample space* and *event*.

A standard die is rolled 10 times and the results are recorded. Particularly Tom was interested in even numbers.

**EXPERIMENT **- the experiment here is rolling a standard die

**SAMPLE SPACE** - the sample space for the experiment is {1,2,3,4,5,6}. That is we could get any of the numbers from 1 to 6 when I roll a standard die.

**EVENT **- the event Tom is interested in is the P(even number). The probability of getting an even number.

Two bags each have $1$1 blue ball and $2$2 yellow balls in them.

A ball is taken from one of the bags without looking. What is the probability that it is a yellow ball?

All the balls from both bags are put into one large new bag and mixed up. What is the probability of randomly picking a yellow ball from the new bag?

A fair die with the numbers $1$1, $6$6, $1$1, $3$3, $2$2 and $1$1 on it is rolled once. What is the probability of rolling a $4$4?

A game in a classroom uses this spinner.

What is the chance of spinning an odd number?

certain

Aeven chance

Bimpossible

Clikely

Dcertain

Aeven chance

Bimpossible

Clikely

DWhat is the chance of spinning a $2$2?

likely

Aimpossible

Bcertain

Ceven chance

Dlikely

Aimpossible

Bcertain

Ceven chance

DWhat is the chance of spinning a number less than $8$8?

likely

Aimpossible

Beven chance

Ccertain

Dlikely

Aimpossible

Beven chance

Ccertain

D