Calculating Probabilities & Events (Edexcel A Level Maths: Statistics)

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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.

3-1-1-fig1-sample-space

    • 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
  • 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
      • (You will have seen this written as A intersection B at GCSE)
    • A or B  means event  happens, or event  happens, or both happen
      • (You will have seen this written as begin mathsize 16px style A union B end style at GCSE)
    • not A means the event A does not happen
      • (You will have seen this written as A' at GCSE)
  • Notation – the way probabilities are written is formal and consistent at A-level
    • begin mathsize 16px style P left parenthesis A right parenthesis equals 0.6 end style           “the probability of event A  happening is 0.6”
    • begin mathsize 16px style P left parenthesis A apostrophe right parenthesis equals 0.4 end style          “the probability of event  A not happening equals 0.4”

(This is sometimes written as begin mathsize 16px style P left parenthesis A with bar on top right parenthesis end style)

    • P left parenthesis X less or equal than 4 right parenthesis equals 0.4      “the probability of X being less than four is 0.4”

How do I solve A level probability questions?

  •  The big difference with probability at A level is the language and the notation used
  • Recall basic results of probability
    • straight P left parenthesis " success " right parenthesis equals fraction numerator number space of space ways space to space get space " success " over denominator total space number space of space outcomes end fraction
      • It is important to understand that the above only applies if all outcomes are equally likely
    • straight P left parenthesis not space straight A right parenthesis equals 1 minus straight P left parenthesis straight A right parenthesis
      • The probability of “n o t space A” 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

  • Ensure you can interpret common ways of displaying data – from frequency tables, histograms, box plots and other ways to illustrate data
    • See Revision Notes       2.1.2 Frequency Tables
                                               2.2.1 Data Presentation
                                               2.2.2 Box Plots & Cumulative Frequency
                                               2.2.3 Histograms
    • Be particularly careful when using histograms
      • These use frequency density, not frequency
      • Using parts of bars may be required due to where class boundaries fall so values will be estimates (using the proportion of the bar needed, sometimes called interpolation)

Worked example

100 skydivers took part in an all-day charity event, with the altitude of the aeroplane at which they jumped from summarised in the histogram below.

3-1-1-fig2-we-diagram

(a)
Use the histogram to find the probability that a randomly chosen skydiver jumped from the aeroplane at an altitude

 

(i)
between 14 000 and 16 000 feet,
(ii)
between 16 000 and 20 000 feet.

 

(b)
Estimate the probability that a randomly chosen skydiver jumped from the aeroplane at an altitude between 13 000 and 15 000 feet.

1x0Cpl4~_3-1-1-fig2-we-solution-part-1

3-1-1-fig2-we-solution-part-2

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)

  • For two independent events, A and B

straight P left parenthesis A space bold AND space B right parenthesis equals straight P left parenthesis A right parenthesis cross times straight P left parenthesis B right parenthesis

e.g.     straight P left parenthesis " 6 space on space straight a space dice " space bold AND bold space " heads space on space straight a space coin " right parenthesis equals 1 over 6 cross times 1 half equals 1 over 12            

  • Independent events could refer to events from different experiments

What are mutually exclusive events?

  • Mutually exclusive events cannot occur simultaneously
    • straight P left parenthesis A space AND space B right parenthesis equals 0
  • 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

straight P left parenthesis A space bold OR space B right parenthesis equals straight P left parenthesis A right parenthesis plus straight P left parenthesis B right parenthesis

e.g.  straight P left parenthesis " 6 space on space straight a space dice " space bold OR bold space " 5 space on space straight a space dice " right parenthesis equals 1 over 6 plus 1 over 6 equals 2 over 6 space space space space open parentheses 1 third close parentheses               

  • 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 cross times
    • mutual exclusivity is OR and is plus
  • 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 3.1.2 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

Just for fun …

  • A well-known sports TV broadcaster used to advertise their football matches as either “Live and exclusive” or “Exclusively live” – can you tell the difference?
  • “Live and exclusive” meant that the broadcaster was airing the football match live and was the only broadcaster allowed to air any of the match at any time.
    “Exclusively live” meant that the broadcaster was the only one airing the football match live, but other broadcasters would be able to air any of the match afterwards.

Worked example

(a)
Two events, Q and R are such that straight P left parenthesis Q right parenthesis equals 0.8 and straight P left parenthesis Q space a n d space R right parenthesis space equals space 0.1.
Given that Q and R are independent, find straight P left parenthesis R right parenthesis

 

(b)
Two events, S and T are such that straight P left parenthesis S right parenthesis space equals space 2 straight P left parenthesis T right parenthesis .
Given that S and T are mutually exclusive and that straight P left parenthesis S space o r space T right parenthesis equals 0.6 find straight P left parenthesis S right parenthesis and straight P left parenthesis T right parenthesis.

 

(c)
A fair five-sided spinner has sides labelled 2, 3, 5, 7, 11.
Find the probability that the spinner lands on a number greater than 5.

3-1-1-fig3-we2-solution

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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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.