 Describing Chance

Lesson What is probability?

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.

Language of Probability

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. Experiment or Trial

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.

Sample Space

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 { }.

Event

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.

Example

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.

Worked Examples

QUESTION 1

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

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

2. 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?

QUESTION 2

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?

QUESTION 3

A game in a classroom uses this spinner. 1. What is the chance of spinning an odd number?

certain

A

even chance

B

impossible

C

likely

D

certain

A

even chance

B

impossible

C

likely

D
2. What is the chance of spinning a $2$2?

likely

A

impossible

B

certain

C

even chance

D

likely

A

impossible

B

certain

C

even chance

D
3. What is the chance of spinning a number less than $8$8?

likely

A

impossible

B

even chance

C

certain

D

likely

A

impossible

B

even chance

C

certain

D

Outcomes

S4-3

Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence