Probability Formulae

What is meant by probability formulae?

If you have seen any of the other Revision Notes on probability, almost every one contains a formula
- These however are often disguised in Venn diagrams, tree diagrams etc
- Sometimes a diagram can be easier to use than a formula
This Revision Note rounds up all these formulae and introduces one or two new ones
You are not expected to use these formulae and none are given in the formulae booklet but they can help solve problems

What probability formulae are there?

Some formulae only apply under certain conditions
The formulae for probability you should’ve encountered so far
- for independent events, $P (A \cap B) = P (A) \times P (B)$ (“AND”, intersection)
- for mutually exclusive events, $P (A \cup B) = P (A) + P (B)$ (“OR”, union)
- $P (A') = 1 - P (A)$ (“NOT”, complement/prime/dash)
- $P (A | B) = \frac{P (A \cap B)}{P (B)}$ or $P (A \cap B) = P (B) \times P (A | B)$
  (“GIVEN THAT”)
  This second version in particular is referred to as the MULTIPLICATION FORMULA (see diagram below)
- For independent events, $P (A | B) = P (A)$

3-2-4-fig1-venn-tree-add-multiply-part-1

~-vnCKhG_3-2-4-fig1-venn-tree-add-multiply-part-2

The addition formula can be rearranged in two very similar ways so be careful
- $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ and

$P (A \cap B) = P (A) + P (B) - P (A \cup B)$

- The special case is when A and B are mutually exclusive; in such cases then

$P (A \cap B) = 0$ and so $P (A \cup B) = P (A) + P (B)$

How do I solve problems using probability formulae?

This is a combination of
- Recognising the set notation used
- Taking note of any independent or mutually exclusive events
- Using an appropriate diagram (Venn, mini-Venn, tree, two-way table)
- Converting worded questions into AND, OR, GIVEN THAT etc statements
- Knowing when a formula could be used
In the majority of cases a diagram can be used – so if you do not like using the formulae or find them confusing
- For events that happen in succession, use a tree diagram; Venn diagrams suit most other purposes but two-way tables can be easier to follow

Worked example

In the fictional World Stare Out Championships players compete by staring at each other with the player blinking first losing. A match cannot be drawn.

During each day of the championships three matches are scheduled to take place but if the first two matches both take more than an hour each, then the third match cannot take place that day.

A statistician notices from past fictional championship records that the probability of the first match taking longer than an hour is 0.15 and that the probability of only two matches taking place on any day is 0.06. They also notice that the probability that at least one of the first two matches takes longer than an hour is 0.32.

Find the probability that

(i)

the second match of a day takes longer than an hour to complete

(ii)

the first match takes longer than an hour given that the second match takes longer than an hour

wov7lfVN_3-2-4-fig2-we-solution-part-1

3-2-4-fig2-we-solution-part-2

Examiner Tip

If in any doubt always start with a diagram
- a Venn diagram can be used for most problems
- a two-way table can be easier to read if it’s possible to construct one
- a tree diagram is useful if you are looking at an event that follows another
Remember that all probability formulae are given in the formulae booklet but not necessarily in the most user-friendly way; a quick look could just be enough to jog your memory though!

Probability Formulae (CIE A Level Maths: Probability & Statistics 1)

Revision Note