How to describe probabilities and the probability scale - KS3 Maths

Key points

A series of two written events. Each event is labelled, has an image, and is surrounded by a curved rectangular box. Event A. A yellow counter is taken from a bag that contains one hundred yellow and two blue counters. Event B. A fair six sided die is rolled to give a number greater than two. The image for event A is a yellow and blue counter. The image for event B is a standard die. — Image caption,
Probability is a measure of the likelihood of events happening.

is the chance of an event happening. It is expressed as a value on a scale from 0 to 1. It can be written as a fraction, a decimal or a percentage.
Where an is impossible, its probability is 0 and where it is certain its probability is 1. This can be presented visually on a .
A probability scale allows the probability of events to be placed in order of . The probabilities marked on the probability scale may be used to make statements comparing different outcomes.
The vocabulary used in probability includes impossible, unlikely, even chance, likely and certain. These descriptions can help order the likelihood of a set of events.

Describing probabilities

The probability (chance) of something happening can be described using particular words:

Certain
Very likely
Likely
Even chance
Unlikely
Very unlikely
Impossible

The for writing the probability of an event is P(event).
The is the set of all possible outcomes. This can be written as a list, S = {……}.
- For example if a fair coin is tossed the possible are {H, T} for heads or tails. The probability of getting a head is written as P(H).
The greater the proportion of times an event can happen the greater (or more likely) the probability. Events can be ordered by the probability of them happening (how likely each event is).

Examples

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The probability of the spinner landing on blue can be described in words.
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The probability of the spinner landing on blue on spinner A is described as impossible because there are no blue sectors. The spinner cannot land on blue. The probability of the spinner landing on blue on spinner B is described as certain because all the sectors are blue. The spinner must land on blue.
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The probability of the spinner landing on blue on spinner C is described as unlikely because there are fewer blue sectors than white sectors. The probability of the spinner landing on blue on spinner D is described as likely because there are more blue than white sectors.
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The probability of the spinner landing on blue on spinner E is described as an even chance because there are an equal number of blue and white sectors. The probability of landing on blue is equal to the probability of landing on white.
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The probability of the spinner landing on blue on spinner D is described as likely. There are more blue than white sectors. The probability of the spinner landing on blue on spinner F is described as very likely. There are even more blue sectors on spinner F than on spinner D.
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P(event) is the convention for writing ‘the probability of an event’. P(blue) means ‘the probability that the outcome is blue’. S = {…} is the sample space, listing the possible outcomes. Where the spinners can show blue or white there are two outcomes in the sample space. There is no indication of how likely the outcomes are.
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Order these events by their probability, from least likely to most likely: rolling an odd number, rolling a number less than 3, rolling a 3, rolling a number greater than 6, and rolling a number less than 7
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A fair die is rolled. The sample space is the numbers 1, 2, 3, 4, 5 and 6. These are all the possible outcomes. The probability of each event can be described in words: Impossible – Very unlikely – Unlikely – Even chance – Likely – Very likely – Certain.
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P(odd) has an even chance because there are an equal number of odd and even numbers. P(< 3) is unlikely because only two of the six numbers are less than 3. P(3) is very unlikely as only one of the six numbers is a three. P(> 6) is impossible because none of the numbers on a die are greater than 6. P(< 7) is certain because all of the numbers on a die are less than 7. The probabilities of the events can now be ordered.
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In order of likelihood, from impossible to certain, the probabilities are P(> 6), P(3), P(< 3), P(odd) and P(< 7).

Practise sample spaces and outcomes

Practise describing different probabilities with this quick quiz.

Understanding the probability scale

The probability scale can be labelled in fractions, decimals or percentages.
The scale goes from 0 to 1 (0% to 100%):
- An impossible has a probability of zero (0%)
- A certain event has a probability of 1 (100%)
- Other events have probabilities between 0 and 1
The probabilities of events can be compared using their positions on the .
The probability scale can be subdivided into equal intervals to show other probabilities:
- For example, four intervals labelled from 0 to 1 with subdivisions marked ¼ and ½ and ¾. This is the same as 0% and 100% with subdivisions marked 25%, 50%, and 75%.

Examples

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The probability scale is labelled from 0 to 1. An impossible event has a probability of 0. A certain event has a probability of 1. Other events have probabilities between 0 and 1
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The probabilities of events can be compared using their positions on the probability scale. Each letter represents the probability of taking a blue ball at random from one of the bags.
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Event A shows that the probability of taking a blue ball at random from a bag is zero. It is impossible. The bag on the far right has no blue balls. P(Blue) = 0
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Event B shows that the probability of taking a blue ball at random from a bag is ¼ (25%). This means that a quarter of the balls in the bag are blue. This is true for the bag on the far left, one of the four balls is blue. P(Blue) = ¼.
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Event C shows that the probability of taking a blue ball at random from a bag is ½ (50%). This means that half of the balls in the bag are blue. This is true for the bag that contains two blue and two white balls, half the balls are blue. P(Blue) = ½.
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Event D shows that the probability of taking a blue ball at random from a bag is 1 (100%). This means that the probability of picking a blue ball is certain. This is true for the bag that contains only blue balls. All the balls are blue. P(Blue) = 1
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A bag contains counters. The counters are orange, white, or blue. The probability of taking a counter of a particular colour from the bag is shown on the probability scale. Statements can be made based on this information.
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The probability of taking a blue counter out of the bag is greater than the probability of taking a white counter. There are more blue counters than white counters in the bag. The number of counters is not known.
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The probability of taking a white counter out of the bag is greater than the probability of taking an orange counter. There are more white counters than orange counters in the bag. The number of counters is not known.
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The probability of a blue counter is greater than ½, this means that more than half of the counters are blue. The probability of a white counter is ¼, this means that a quarter of the counters are white. There is not enough information to decide how many of each coloured counter is in the bag.

Question

A box of chocolates contains toffees and fudges. The probability scale shows the probability of taking a toffee from the box.

Decide if each statement below is true or false or if there is not enough information to tell.

An image of a horizontal probability scale. The scale has been divided into two parts. The left end of the scale has been marked and labelled as zero. The right end of the scale has been marked and labelled as one. The middle of the scale is marked. Written above this mark: Toffee, with a downwards pointing arrow. Drawn left: A box of chocolates. Written below are three statements. Statement one: there are five toffees. Statement two: Half of the chocolates are not toffees. Statement three: There are more toffees than any other type of chocolate. The arrow is coloured orange.

Practise describing probabilities

Practise describing probability and using the probability scale with this quiz. You may need a pen and paper to help you with your answers.

Quiz

Real-life maths

An image of a raffle ticket being drawn.

Charities often raise money by having raffles. In a raffle, tickets are selected by lucky-dip. Usually, tickets that end in a zero or a five win a prize, which means one ticket in every five will win a prize.

Charities need to calculate how many tickets they need to sell and how much prizes cost in order to make a profit for the charity. Even though the chance of winning is unlikely, raising money for charity and the bonus of a possible prize makes it worthwhile.

Other raffle ticket prizes may be selected during the event and are not based on the ticket number. The chance of winning one of these prizes is less as there are fewer prizes for more tickets.