Maths

Probability

Probability is about estimating how likely (probable) something is to happen. Probability can be used to predict, for example, the outcome when throwing a die or tossing a coin.

# Using words to describe probabilities

We often use words to describe how probable we think it is that an event will take place. For example, we might say that it is likely to be sunny tomorrow, or that it is unlikely to snow in August.

Have a look at the statements below, and decide whether the following events are: certain, likely, very likely, evens (neither likely or unlikely), unlikely, very unlikely or impossible.

• You buy a lottery ticket and win the jackpot.
• You toss a coin and get heads.
• Christmas will fall on 25 December this year.
• You grow another nose.
• It will rain in the first week of December.
Question

A game is played where 7 beads with digits from 1 to 7 on them are placed in a bag, a bead is taken out, and you then have to guess whether the number on the next one to be taken out will be higher or lower, and so on.

In each case, choose an appropriate word from the following list to complete each sentence:

1. Certain
2. Likely
3. Very likely
4. Unlikely
5. Very unlikely
6. Impossible

Remember that there is only one of each number.

1. If the 1st bead is 7, what is the chance of the 2nd bead being lower than 7?
2. If the 2nd bead is 1, what is the chance of the 3rd bead being lower than 2?
3. If the 3rd bead is 6, what is the chance of the next bead being higher than 4?

• 1. The first bead is 7.
• What is the chance of the next bead being lower than 7?
• The numbers remaining are: 6, 5, 4, 3, 2 and 1.
• So the chance of the 2nd bead being less than 7 is
• certain.
• 2. The 2nd bead is 1.
• The numbers remaining are: 2, 3, 4, 5 and 6.
• The chance of the next bead being lower than 2 is impossible.
• 3. The 3rd bead is 6.
• The numbers remaining are: 2, 3, 4 and 5.
• The chance of the next bead being higher than 4 is unlikely.

# Basic probabilities

Words like 'certain', 'likely', and so on, may not mean the same to everyone. We need to be more precise about how likely something (an outcome) is to happen. The probability of an outcome can have any value between 0 (impossible) and 1 (certain). It may be a fraction, decimal or percentage.

When different outcomes of an event are equally likely (for example, getting a head when you toss a coin), you can use a formula to calculate the probability of outcomes.

When you throw a fair dice there are six possible outcomes: 1, 2, 3, 4, 5 or 6. There are three ways of getting an odd number (1, 3 or 5).

So the probability of getting an odd number =

the number of ways of getting an odd number ÷ total number of possible outcomes = 3/6 = 1/2 or 0.5, or 50%.

## Activity

Click to play the probability activity

## Events and probabilities

The table below gives some examples of events and how their probability can be calculated.

EventOutcomeNumber of ways to get this outcomeTotal number of possible outcomesProbability of outcome
Throwing a fair, 6-sided diceGetting an odd number 3 63/6
Throwing a fair coinGetting 'tails'12 50%
Choosing a playing card from a full pack without lookingThe suit being spades 13 5213/52
Choosing a playing card from a full pack without lookingThe card being a 'ten' 4 52 4/52
Throwing a fair, 6-sided diceGetting a number less than 5 4 64/6

You may sometimes need to list all the possible outcomes of an event. The key is to work systematically - do not just list all the outcomes randomly. Here is an example:

Question

Imagine that you had to find all the different orders in which three people (Anita, Benita and Carol) could finish in a race.

The first step (to save you writing too much) is to label the people A, B and C - but make sure you mention this. Jot down the different orders in which Anita, Benita and Carol could finish.

ABC, ACB, BCA, BAC, CAB, CBA

Make sure you can 'see' the pattern in the order of choices.

# Calculating probabilities

Here are two fair spinners. The total score is the sum of the two numbers the arrows point to.

Jot down, systematically, all the possible outcomes for the two spinners. You will find it useful to use a table of results, as shown.

Triangular spinnerSquare spinnerTotal score
101
112
123
134
202
213
224
235
303
314
325
336

Use the table to answer these questions:

Question

How many different possible outcomes are there?

12. All you have to do is count the number of entries in the table.

Question

How many outcomes gave a total score of 2?

Only 2 outcomes give a total score of 2: a '1' on each spinner, or a '2' on the triangular spinner and '0' on the square one.

Question

What is the probability of getting a total score of 2?

The number of outcomes giving a total score of 2 ÷ the total number of outcomes = 2/12 or 1/6

Question

How many outcomes gave a total score of 4?

3.

Question

What is the probability of getting a total score of 4?

The number of outcomes giving a total score of 4 ÷ the total number of outcomes = 3/12 or 1/4

## Relative frequency

You can estimate probabilities from an experiment. These are sometimes called experimental probabilities.

For example, in an experiment where you drop a drawing pin:

• The pin lands up 279 times.
• The pin lands down 721 times.
• The total number of throws is 1000.

So the probability of the drawing pin landing up is:

The number of times this outcome occurs (pin up) ÷ total number of outcomes (or trials) = 279/1000 (or 0.279, or 27.9 %).

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