Conditional Probability

Topic Overview

During this revision guide, you will be introduced to the concept of 'conditional probability'.

As mentioned in the revision guide "Probability Diagrams" probability related problems are asking you to estimate to likelihood of an event occurring. There are several terms used to describe probability, including; 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).

Conditional probability is the term used to describe the probability of an event occurring wherein the outcome is dependent upon a previous event.

For example, if a number of beads are being taken out a bag. If there are three red beads and four green beads in a bag, then the probability of choosing a red bead will be 3/7 and the probability of choosing a green bead will be 4/7 .

However, if you then must pick another bead out of the bag without replacing the first bead your chose, then the probability of choosing the second bead will be conditional upon the first bead you chose.

For instance, if the first bead you chose was green, then the probability of choosing another green bead will be 3/6 and the probability of choosing a red bead will be 3/6.

Alternatively, if the first bead you chose was red, then the probability of choosing a green bead will be 4/6 and the probability of choosing another red bead will be 2/6.

Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems related to conditional probabilities. 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:

  • Recognise and calculate conditional probabilities
  • List all outcomes for single events, and for two successive events, in a systematic way and derive relative probabilities
  • Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1

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

Worked Examples

During your GCSE maths exam, you will be required to calculate the likelihood of various conditional probabilities:

1 - Representing conditional probabilities on tree diagrams

As mentioned in the revision guide "Probability Diagrams"; tree diagrams are used to demonstrate all of the possible outcomes of an event. According to the rules of tree diagrams:

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

- In a tree diagram, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.

When calculating conditional probability problems, it is important to remember that, after each event the numerators and denominators of each individual outcome must change due to the fact that one object has been removed and not replaced:

Example

(a) A bag contains 2 blue beads and 3 red beads.

(i) Create a tree diagram which demonstrates the probabilities if one bead is chosen from the bag and then a second bead is chosen from the bag without the first bead being replaced.

(ii) Use your tree diagram to calculate the probability of the first bead being blue and the second bead being red.

Solution

(a) (i) There are a total of 5 beads in the bag; of which 2 are blue and 3 are red.

Therefore, the probability of the first bead being blue is 2/5 and the probability of the first bead being red is 3/5.

Now that one bead has been taken out of the bag, there are only 4 left remaining.

If the first bead taken out was blue, then there would be 1 blue bead and 3 red beads remaining. Therefore, the probability of the second bead being blue is 1/4 and the probability of the second bead being red is 3/4.

Alternatively, if the first bead taken out was red, then there would be 2 blue beads and 2 red beads remaining. Therefore, the probability of the second bead being blue is 2/4 and the probability of the second bead being red is 2/4.

This information can be represented in the following tree diagram:

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(ii) Using, your tree diagram, you can multiply along the branches to calculate the probability of the first bead being blue and the second bead being red:

(Probability of the first bead being blue) x (Probability of the second bead being red) =

\[\frac{2}{5}*\frac{3}{4}=\frac{6}{20}=\frac{3}{10}\]

Therefore, the probability of the first bead being blue and the second bead being red is

\[\frac{3}{10}\]

2 - Conditional probability notation

When answering conditional probability related problems during your GCSE maths exam, you will encounter various probability notations. These notations are listed below:

P(A) : this notation means "the probability of event A occurring"

For example, the probability of choosing a green ball from a bag is 4/7

P(B) : this notation means "the probability of event B occurring"

For example, the probability of choosing a red ball from a bag is 3/7

The symbol "|" is used to mean "given". Therefore, the notation P(B|A) means "the probability of event B occurring given that event A has already occurred". As a result, the notation P(B|A) is often referred to as the "Conditional Probability of B given A"

For example, the probability of the second ball being green would be 3/6 and the probability of the second ball being red would be 3/6 , because there will be one less green ball in the bag.

P(A and B) means the probability of event A occurring and then the probability of event B occurring given that event A has already occurred. This can be written as:

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

For example, the probability of both the first and second ball being green would be:

\[\frac{4}{7}*\frac{3}{6}=\frac{12}{42}=\frac{2}{7}\]

This expression can be rearranged so that: P(B|A) = P(A and B) ÷ P(A).

It is often useful to rearrange these notation forms if you are asked to calculate the probability of certain outcomes where there are unknown values.

Example

(a) A crate contains 6 oranges and 8 apples. Two pieces of fruit are chosen at random from the box without replacement.

What is the probability that the pieces of fruit chosen are different types of fruit?

Solution

(a) Although it is advisable that you use a tree diagram to solve probability questions, it is possible to solve this question solely using notations:

Using the information given in the question, you know that there are a total of two possible outcomes wherein the two pieces of fruit chosen are different types:

Outcome Number One: Orange followed by Apple

Using notation, you can state that "Event A" would be choosing an orange first and that "Event B" would be choosing an apple second. In notation form, the probability of this outcome can be written as:

\[P\left( A \right)=\frac{6}{14}\] \[P\left( B|A \right)=\frac{8}{13}\]

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

\[\frac{6}{14}*\frac{8}{13}=\frac{48}{182}=\frac{24}{91}\]

Outcome Number Two: Apple followed by Orange

Using notation, you can state that "Event C" would be choosing an apple first and that "Event D" would be choosing an orange second. In notation form, the probability of this outcome can be written as:

\[P\left( C \right)=\frac{8}{14}\] \[P\left( D|C \right)=\frac{6}{13}\]

Therefore P(C and D) = P(C) x P(D|C) =

\[\frac{8}{14}*\frac{6}{13}=\frac{48}{182}=\frac{24}{91}\]

As a result, you can calculate that the probability of choosing two different types of fruit would be

\[\frac{24}{91}+\frac{24}{91}=\frac{48}{91}\]

Exam Tips

  1. When calculating conditional probability problems, it is important to remember that, after each event, the numerators and denominators of each individual outcome must change because one object has been removed and not replaced.
  2. When using a tree diagram, the sum of the probabilities for any set of branches in a tree diagram will always be 1.
  3. Remember that, in a tree diagram, you can calculate the probability of an outcome by multiplying along the branches and adding vertically.
  4. Write down ALL of your working out when calculating probability outcomes.
  5. Memorise the following probability notations: P(A) means "the probability of event A occurring" and P(B) means "the probability of event B occurring".
  6. Memorise that the symbol "|" is used to mean "given". Therefore, the notation P(B|A) means "the probability of event B occurring given that event A has already occurred".
  7. Remember that the notation P(A and B) means the probability of event A occurring and then the probability of event B occurring given that event A has already occurred. This can be written as: P(A and B) = P(A) x P(B|A). This expression can be rearranged so that: P(B|A) = P(A and B) ÷ P(A).

Topic Summary

Conditional probability questions can appear complex at first but, once you have mapped out all of the possible outcomes of various events occurring, they are relatively easy to solve. For this reason, it is recommended that you always construct a tree diagram when answering conditionally probability outcomes; even if you have not been instructed to do so. A tree diagram enables you to logically follow the probability of various events occurring by visually observing the path of the branches on your tree diagram.

As long as you remember that both the numerator and denominator of probabilities must change after each event occurs, then you will be able to correctly calculate the conditional probabilities of any outcomes. Remember to write down ALL of your working out in order to score maximum methods marks, as well as adding together all of the probabilities on your tree diagram. If all of your possible outcomes add up to 1, then you know you have calculated your conditional probabilities correctly.

Ultimately, by tackling as many past paper questions as possible you can consolidate your knowledge of a wide range of probability and conditional probability related questions in order to score highly during your actual GCSE maths exam.

Related Topics

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