Conditional Probability

What is conditional probability?

Conditional probability refers to situations where the probability of an event changes or is dependent on other events having already happened
For example, drawing names from a hat, without replacement
- If there are 10 (different) names in a hat to start with
- the first name drawn has the probability of $\frac{1}{10}$ of being a particular name
- the second name drawn has probability $\frac{1}{9}$ of being a particular name
  - or, if this particular name was the first one to be drawn, it would have probability 0 of being drawn second
- The probability has changed depending on what has happened already
Conditional probabilities often occur in the context of Venn diagrams, tree diagrams or two-way tables
- however questions may also be given in words only
- in such cases it may sometimes be easier to understand what is happening by drawing one of these diagrams
- unless a question tells you to though, drawing a diagram is not essential
- for many questions it is quicker simply to consider the possible options without drawing a diagram
Conditional probability questions are often in the form of "given that" questions
- e.g. Find the probability it will rain today given that it rained yesterday
  - It makes sense that whether or not it rained yesterday would affect the probability of whether or not it rains today
  - The phrase "given that" is not always used in conditional probability questions
  - Like AND/OR, you will need to interpret the phrases used in questions
Conditional probabilities are sometimes written using the 'straight bar' notation $P (A | B)$
- That is read as 'the probability of A given B'
- For example $P (passes | no revision)$ would indicate the probability that a student passes his exams, given that he has done no revision
- That probability is likely to be quite different from $P (passes | lots of revision)$ !

Worked example

A box contains 3 blue counters and 8 red counters.
A counter is taken at random and its colour noted.
The counter is then set aside and not put back into the box.
A second counter is then taken at random, and its colour noted.

Write down the probability that

the second counter is red, given that the first counter was red

ii)

the second counter is blue, given that the first counter was red

iii)

the second counter is red, given that the first counter was blue

iv)

the second counter is blue, given that the first counter was blue.

If the first counter was red, then only 7 red counters remain in the box.
There are still 3 blue counters, and 10 counters in total.

$\begin{array}{rcl} \frac{7}{10} \end{array}$

ii)

If the first counter was red, then only 7 red counters remain in the box.
There are still 3 blue counters, and 10 counters in total.

$\begin{array}{rcl} \frac{3}{10} \end{array}$

iii)

If the first counter was blue, then only 2 blue counters remain in the box.
There are still 8 red counters, and 10 counters in total.

$\frac{8}{10} (= \frac{4}{5})$

iv)

If the first counter was blue, then only 2 blue counters remain in the box.
There are still 8 red counters, and 10 counters in total.

$\frac{2}{10} (= \frac{1}{5})$

Conditional Probability (AQA GCSE Maths)

Revision Note