Probability Basics
What do I need to know about probability for AS and A level Mathematics?
 The language used in probability can be confusing so here are some definitions of commonly misunderstood terms
 An experiment is a repeatable activity that has a result that can be observed or recorded; it is what is happening in a question
 An outcome is the result of an experiment
 All possible outcomes can be shown in a sample space – this may be a list or a table and is particularly useful when it is difficult to envisage all possible outcomes in your head
e.g. The sample space below is for two fair foursided spinners whose outcomes are the product of the sides showing when spun.

 An event is an outcome or a collection of outcomes; it is what we are interested in happening
 Do note how this could be more than one outcome
e.g. For the spinners above,
the event “the product is 2” has one outcome but
the event “the product is negative” has 6 outcomes
 Do note how this could be more than one outcome
 An event is an outcome or a collection of outcomes; it is what we are interested in happening
 Terminology  be careful with the words 'not', 'and' and 'or'
 A and B means both the events A and B happen at the same time
 A and B is formally written as (∩ is called intersection)
 A or B means event A happens, or event B happens, or both happen
 A and B is formally written as (∪ is called union)
 not A means the event A does not happen
 not A is formally written as A' (pronounced "A prime")
 A and B means both the events A and B happen at the same time
 Notation – the way probabilities are written is formal and consistent at Alevel
 “the probability of event A happening is 0.6”
 “the probability of event A not happening equals 0.4”
(This is sometimes written as )

 “the probability of being less than four is 0.4”
How do I solve A level probability questions?
 Recall basic results of probability

 It is important to understand that the above only applies if all outcomes are equally likely

 The probability of “” is the complement of the probability of “A”
 One of the easiest results in probability to understand,
one of the hardest results to spot!

 Be aware of whether you are using theoretical probabilities or probabilities based on the results of several experiments (relative frequency). You may have to compare the two and make a judgement as to whether there is bias in the experiment.
e.g. The outcomes from rolling a fair dice have theoretical probabilities but the outcomes from a football match would be based on previous results between the two teams
 For probabilities based on relative frequency, a large number of experiments usually provides a better estimate of the probability of an event happening
 Frequencies or probabilities may have to be read from basic statistical diagrams such as bar charts, boxandwhisker diagrams, stem and leaf diagrams, etc
Worked Example
A fair, fivesided spinner has its sides labelled 2, 5, 8, 10 and 11.
Find, from one spin, the probability that the spinner shows
(i)
8
(ii)
a prime number
(iii)
an odd prime number
(iv)
a number other than 5.
Exam Tip
 Most probability questions are in context so can be long and wordy; go back and reread the question, several times, whenever you need to
 Try to get immersed in the context of the question to help understand a problem
Independent & Mutually Exclusive Events
What are independent events?
 Independent events do not affect each other
 For two independent events, the probability of one event happening is unaffected by the outcome of the other event
 e.g. The events “rolling a 6 on a dice” and “flipping heads on a coin” are independent
 the outcome “rolling a 6” does not affect the probability of the outcome “heads” (and vice versa)
 e.g. The events “rolling a 6 on a dice” and “flipping heads on a coin” are independent
 For two independent events, A and B
e.g.
 Independent events could refer to events from different experiments
What are mutually exclusive events?
 Mutually exclusive events cannot occur simultaneously
 For two mutually exclusive events, the outcome of one event means the other event cannot occur
 e.g. The events “rolling a 5 on a die” and “rolling a 6 on a die” are mutually exclusive
 For two mutually exclusive events, A and B
e.g.
 Mutually exclusive events generally refer to events from the same (single trial of an) experiment
 Mutually exclusive events cannot be independent; the outcome of one event means the probability of the other event is zero
How do I solve problems involving independent and mutually exclusive events?
 Make sure you know the statistical terms – independent and mutually exclusive
 Remember
 independence is AND(∩) and is
 mutual exclusivity is OR (∪) and is
 Solving problems will require interpreting the information given and the application of the appropriate formula
 Information may be explained in words or by diagram(s)
(including Venn diagrams – see Revision Note Venn Diagrams)
 Showing or determining whether two events are independent or mutually exclusive are also common
 To do this you would show the relevant formula is true
Worked Example
(a)
Two events, and are such that and .
Given that and are independent, find
Given that and are independent, find
(b)
Two events, and are such that .
Given that and are mutually exclusive and that find and .
Given that and are mutually exclusive and that find and .
(c)
A fair fivesided spinner has sides labelled 2, 3, 5, 7, 11.
Find the probability that the spinner lands on a number greater than 5.
Find the probability that the spinner lands on a number greater than 5.
(a)
Two events, and are such that and .
Given that and are independent, find
Given that and are independent, find
(b)
Two events, and are such that .
Given that and are mutually exclusive and that find and .
Given that and are mutually exclusive and that find and .
(c)
A fair fivesided spinner has sides labelled 2, 3, 5, 7, 11.
Find the probability that the spinner lands on a number greater than 5.
Find the probability that the spinner lands on a number greater than 5.
Exam Tip
 Try to rephrase questions in your head in terms of AND and/or OR !
e.g. A fair sixsided die is rolled and a fair coin is flipped.
“Find the probability of obtaining a prime number with heads.”
would be
“Find the probability of rolling a 2 OR a 3 OR a 5 AND heads.”