Probability Diagrams

Topic Overview

When solving probability problems, you are being asked to estimate how likely an event is to happen. For example, during your GCSE maths exam you may be asked to predict the outcome when a dice is rolled or a coin is tossed.

When answering these questions, there are several terms you can use which are specific to the topic of probability. These terms include; certain (i.e. the incident will definitely occur), likely, very likely, even (i.e. it is neither likely nor unlikely for the incident to occur), unlikely, very unlikely or impossible (i.e. the incident will never occur).

There are two different types of probability event: 'independent' and 'dependent'.

If two events are independent, then the outcome of one event does not affect the outcome of the other event. For example, if you toss two coins, then the probability a one coin turning up heads will not affect the probability of the other coin turning up heads.

However, if two events are dependent, then the outcome of one event will affect the outcome of the other event. For example, if you have 3 beads in a bag of which 3 are blue and 3 are red, then each time you remove a bead from the bag you will change the probability of selecting the coloured beads which are left in the bag.

As well as these probability terms, this revision guide will demonstrate several examples of probability diagrams which you can use in order to display the likelihood that certain incidents will occur.

Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems related to probability. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Understand and use the vocabulary of probability and probability scale
  • Understand and use estimates or measures of probability
  • Design and use two-way tables for various data
  • List all of the outcomes for single events, and for two successive events, in a systematic way and derive relative probabilities
  • Use tree diagrams to represent outcomes of events

Listed below are a series of summaries and worked examples to help you solidify your knowledge about probability and probability diagrams.

Worked Examples

1 - Using tree diagrams to calculate probabilities

'Tree diagrams' are used to demonstrate all of the possible outcomes of an event and can be used to calculate the probability of these events occurring. In a tree diagram, each branch represents a possible outcome and is accompanied by the probability of this outcome occurring. It is important to remember that:

The sum of the probabilities for any set of branches in a tree diagram will always be 1.

It is also important to note that, in a tree diagram, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.

In order to calculate the probability of an event occurring, you can use the following rule:

Probability = Number of ways an event can happen ÷ Total number of outcomes

Once you have calculated this information, you can display these probabilities via a tree diagram:

Example

(a) (i) Create a tree diagram which displays the probabilities of tossing a coin three times.

(ii) Using this tree diagram, calculate the probability of tossing a coin and turning up heads three times in a row.

(iii) Using this tree diagram, calculate the probability of tossing a coin and turning up heads two times.

Solution

(a) (i) When you toss a coin, there are two possible events which can occur; either the coin will turn up heads or the coin will turn up tails.

The likelihood of each event occurring is: 1/2 , 50%, or 0.5 .

Each time you toss the coin, the likelihood of these events occurring will remain the same. This information can be demonstrated by the following tree diagram:

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(ii) From this diagram, it is evident that there are eight possible outcomes:

Heads, Heads, Heads (HHH) Heads, Heads, Tails (HHT) Heads, Tails, Heads (HTH) Heads, Tails, Tails (HTT) Tails, Heads, Heads (THH) Tails, Heads, Tails (THT) Tails, Tails, Heads (TTH) Tails, Tails, Tails (TTT)

The question has asked you to calculate the probability of turning up heads three times in a row, i.e. Heads, Heads, Heads (HHH).

In order to calculate the probability of this outcome, you need to look at its associated branches. As mentioned earlier, you can calculate the probability of an outcome by multiplying along the branches. Therefore:

The probability of turning up heads three times in a row =

\[\frac{1}{2}*\frac{1}{2}*\frac{1}{2}=\frac{1}{8}\]

(iii) You have also been asked to calculate the probability of tossing a coin and turning up heads two times.

Looking at the list of possible outcomes, there are three separate outcomes where you could turn up heads two times:

Heads, Heads, Tails (HHT) Heads, Tails, Heads (HTH) Tails, Heads, Heads (THH)

As mentioned earlier, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.

Therefore, the probability of turning up heads two times is:

\[(\frac{1}{2}*\frac{1}{2}*\frac{1}{2})+(\frac{1}{2}*\frac{1}{2}*\frac{1}{2})+(\frac{1}{2}*\frac{1}{2}*\frac{1}{2})=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\]
\[=\frac{3}{8}\]

2 - Using Venn diagrams to calculate probabilities

A 'Venn diagram' is another useful way to represent mathematical or logical sets of information. In a Venn diagram, the position and overlapping of circles are used to indicate the relationships between different sets of information.

For example, a Venn diagram can be used to display how many people own a cat 'A', how many people own a dog 'B', how many people own both a cat and a dog 'A and B' and how many people own neither a cat nor a dog 'not A and not B':

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There are several terms used to describe the different sets within Venn diagrams:

A U B or "A union B" refers to everything which is included in either of the sets. For example, in the following diagram, the answers are 1, 2, and 3.

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A ∩ B or "A intersect B" refers only to the values which are in both of the sets. For example, in the following diagram, the answer is 2.

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Ac or ~A or "not A" refers to everything outside of the region of A. For example, in the diagram below, the answers are 3 and 4.

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A – B or "A minus B" refers to everything in the region of A, with the exception of anything which is in its overlap with B. For example, in the following diagram the answer is 1.

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~(A U B) or "not (A union B)" refers to everything outside the regions of A and B. For example, in the following diagram the answer is 4.

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~(A ∩ B) or "not (A intersect B)" refers to everything outside of the overlap of A and B. For example, in the following diagram the answers are 1, 3 and 4.

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During your GCSE maths exam, you may be asked to create various Venn diagrams and use them to evaluate various sets of data:

Example

(a) Using the following Venn diagram, identify the values which are in:

(i) the union of A and B

(ii) the intersection of A and B

(iii) the region of A minus B

(iv) the region of not (A union B)

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Solution

(a) (i) The union of A and B refers to everything which is included in either of the sets. Therefore, the values in question are : 1, 3, 5, 6, 7, 8, 9 and 10

(ii) The intersection of A and B refers only to the values which are in both of the sets. Therefore, the values in question are: 7 and 9

(iii) The region of A minus B refers to everything in the region of A, with the exception of anything which is in its overlap with B. Therefore, the values in question are: 1, 3 and 5

(iv) The region of not (A union B) refers to everything outside the regions of A and B. Therefore, the values in question are: 2 and 4

3 - Two-way tables

A 'two-way table' can be used to organize the sets of data for two categorical variables. In a two way table, the values of the row variable run across the table, and the values of the column variable label run down the table. These two way tables are very useful for comparing the relationship between two categorical variables:

Example

(a) A survey is carried out on a class of 25 pupils in order to find out the different mobile devices which each pupil owns. Within the survey, there are four possible answers:

  1. A pupil has a mobile phone and a laptop
  2. A pupil has a mobile phone but not a laptop
  3. A pupil has a laptop but not a mobile phone
  4. A pupil does not have a mobile phone or a laptop

The results of the survey are as follows:

  1. Pupils who have a mobile phone and a laptop: 6
  2. Pupils who have a mobile phone but not a laptop: 12
  3. Pupils who have a laptop but not a mobile phone: 3
  4. Pupils who do not have a mobile phone or a laptop: 4

Represent this information via a two way table

Solution

(a) In a two way table, the values of the row variable run across the table, and the values of the column variable label run down the table.

Therefore, the information from the survey can be represented in a two way diagram like so:

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Now that you have allocated rows and columns for each variable, you can insert your numerical values into your two way table:

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4 - Mutually exclusive events

Two events can also be described as 'mutually exclusive’. This means that these two event cannot occur at the same time. For example, if you roll a dice, it is impossible to 'roll a 5' and 'roll a 2' at the same time.

When two events, (A and B), are independent, this can be written as:

P(A and B) = P(A) x P(B)

Alternatively, when two events, (A and B), are mutually exclusive, this can be written as:

P(A or B) = P(A) + P(B)

These two rules form the AND/OR rule which you will be required to use during your GCSE maths exam:

Example

(a) A bag contains 5 green beads and 4 red beads. A bead is taken from the bag, its colour noted, and it is then replaced. A second bead is then taken from the bag. What is the probability that the two beads are different colours?

Solution

(a) This example is asking you to calculate the probability of the first bead being green AND the second bead being red, OR the probability of the first bead being red AND the second bead being green.

In this example, the first bead is replaced before the second bead is taken out. As a result, the first and second beads are independent. Therefore, you can use the rule:

P(A and B) = P(A) x P(B)

Alternatively, the probability of the first bead being green and the second bead being red OR the first bead being red and the second bead being green are mutually exclusive. Therefore, you can use the rule:

P(A or B) = P(A) + P(B)

There are a total of 9 beads in the bag. Therefore, the probability of choosing a green bead first is 5/9 and the probability of choosing a red bead first is 4/9 .

You can use all of this information to calculate the probability of choosing two beads which are two different colours:

\[(\frac{5}{9}*\frac{4}{9})+(\frac{4}{9}*\frac{5}{9})=(\frac{20}{81})+(\frac{20}{81})\]

Therefore, the probability of choosing two beads which are two different colours =

\[\frac{40}{81}\]

Exam Tips

  1. When using a tree diagram, remember that the sum of the probabilities for any set of branches in a tree diagram will always be 1.
  2. Remember that, in a tree diagram, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.
  3. When calculating the probability of an event occurring, memorise the following rule: Probability = Number of ways an event can happen ÷ Total number of outcomes.
  4. Label all of your probability diagrams clearly and concisely, as well as writing down ALL of your working out when calculating probability outcomes.

Topic Summary

When tackling probability related questions, you need to be certain of the relationships between different events occurring. Above all else, you need to memorise that:

  • If two events are independent then the outcome of one event does not affect the outcome of the other event,
  • If two events are dependent, then the outcome of one event will affect the outcome of the other event,
  • When two events are described as mutually exclusive, this means that these two event cannot occur at the same time

If you have a strong understanding of these concepts of probability, you will be able to clearly and logically draw probability diagrams and calculate the likelihood of different outcomes. Whenever you are faced with a probability question, draw a tree or Venn diagram or a two way table; even if you have not been instructed to do so. By drawing these diagrams, you equip yourself with a clear visual aid which will help you understand seemingly complicated problems.

Furthermore, these diagrams will demonstrate that you understand the fundamental concepts of probability to the examiner who is marking your paper. As a result, you will increase your chances of scoring the maximum amount of method marks for the topic of probability.

Related Topics

  • Sampling
  • Charts and Tables
  • Average and Spread
  • Conditional Probability