Calculating Probabilities & Events (Cambridge (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.

• 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  $A\cap B$ (∩ is called intersection)

  • A or B  means event A happens, or event B happens, or both happen

    • A  or B  is formally written as  $A\cup B$(∪ 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

  • $P\left(A\right)=0.6$           “the probability of event A happening is 0.6”

  • $P(A\text{'})=0.4$          “the probability of event A not happening equals 0.4”

(This is sometimes written as $P\left(\overline{A}\right)$)

• $P(X\le 4)=0.4$      “the probability of being less than four is 0.4”

How do I solve A level probability questions?

• Recall basic results of probability

  • $\mathrm{P}("\mathrm{success}")=\frac{\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{to}\mathrm{get}"\mathrm{success}"}{\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{outcomes}}$

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

  • $\mathrm{P}(A\text{'})=1-\mathrm{P}\left(A\right)$

    • The probability of “$notA$” 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 

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

$\mathrm{P}\left(A\mathbf{AND}B\right)=\mathrm{P}\left(A\right)\times \mathrm{P}\left(B\right)$

e.g.     $\mathrm{P}("6\mathrm{on}\mathrm{a}\mathrm{dice}"\mathbf{AND}"\mathrm{heads}\mathrm{on}\mathrm{a}\mathrm{coin}")=\frac{1}{6}\times \frac{1}{2}=\frac{1}{12}$            

• Independent events could refer to events from different experiments

What are mutually exclusive events?

• Mutually exclusive events cannot occur simultaneously

  • $\mathrm{P}\left(A\mathrm{AND}B\right)=\mathrm{P}(A\cap B)=0$

• 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

$\mathrm{P}\left(A\mathbf{OR}B\right)=P(A\cup B)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)$

e.g.  $\mathrm{P}("6\mathrm{on}\mathrm{a}\mathrm{dice}"\mathbf{OR}"\mathrm{heads}\mathrm{on}\mathrm{a}\mathrm{coin}")=\frac{1}{6}+\frac{1}{2}=\frac{4}{6}=\left(\frac{2}{3}\right)$               

• 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 $\times$

  • 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