Calculating Probabilities & Events (CIE A Level Maths: Probability & Statistics 1)

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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
      • A  and B  is formally written as  A intersection B (∩ is called intersection)
    • A or B  means event A happens, or event B happens, or both happen
      • A  or B  is formally written as  A union B(∪ is called union)
    • not A means the event A does not happen
      • not A is formally written as A' (pronounced "A prime")
  • 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”
    • P left parenthesis A apostrophe right parenthesis equals 0.4          “the probability of event A not happening equals 0.4”

(This is sometimes written as P left parenthesis A with bar on top right parenthesis)

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

How do I solve A level probability questions?

  • 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 A apostrophe right parenthesis equals 1 minus straight P left parenthesis 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

  • 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 

(i)
8
(ii)
a prime number
(iii)
an odd prime number
(iv)
a number other than 5.

2-1-1-cie-fig2-we-solution

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 straight P left parenthesis A intersection 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 P left parenthesis A union B right parenthesis space equals space 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 " heads space on space straight a space coin " right parenthesis equals 1 over 6 plus 1 over 6 equals 2 over 6 space equals 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 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, Q and R are such that straight P left parenthesis Q right parenthesis equals 0.8 and text P end text left parenthesis Q space and space R right parenthesis equals 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 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 a n d 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.

 

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

2-1-1-cie-fig3-we-solution_a

(b)
Two events, S and T are such that straight P left parenthesis S right parenthesis 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.
2-1-1-cie-fig3-we-solution_b
(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.
2-1-1-cie-fig3-we-solution_c

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.