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Calculating Probabilities & Events (CIE A Level Maths: Probability & Statistics 1)
Revision Note
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 four-sided spinners whose outcomes are the product of the sides showing when spun.
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- 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 or 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 A-level
- “the probability of event A happening is 0.6”
- “the probability of event A not happening equals 0.4”
(This is sometimes written as )
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- “the probability of being less than four is 0.4”
How do I solve A level probability questions?
- Recall basic results of probability
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- It is important to understand that the above only applies if all outcomes are equally likely
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- 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!
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- 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, box-and-whisker diagrams, stem and leaf diagrams, etc
Worked example
A fair, five-sided spinner has its sides labelled 2, 5, 8, 10 and 11.
Find, from one spin, the probability that the spinner shows
Examiner Tip
- Most probability questions are in context so can be long and wordy; go back and re-read the question, several times, whenever you need to
- Try to get immersed in the context of the question to help understand a problem
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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
Given that and are independent, find
Given that and are mutually exclusive and that find and .
Find the probability that the spinner lands on a number greater than 5.
Given that and are independent, find
Given that and are mutually exclusive and that find and .
Find the probability that the spinner lands on a number greater than 5.
Examiner Tip
- Try to rephrase questions in your head in terms of AND and/or OR !
e.g. A fair six-sided 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.”
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