Calculating Probabilities & Events (Edexcel International AS Maths: Statistics 1): Revision Note

Calculating probabilities & events

What language is used in probability?

• 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

    • (You will have seen this written as $A\cap B$ at GCSE)

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

    • (You will have seen this written as $A\cup B$ 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

  • $P\left(A\right)=0.6$           “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 $X$ being less than four is 0.4”

How do I find probabilities?

•  The big difference with probability at A level is the language and the notation used

• 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

  • $\mathrm{P}\left(\mathrm{not}\mathrm{A}\right)=1-\mathrm{P}\left(\mathrm{A}\right)$

    • 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

• 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)

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

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

• 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

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

• e.g.  $\mathrm{P}("6\mathrm{on}\mathrm{a}\mathrm{dice}"\mathbf{OR}"5\mathrm{on}\mathrm{a}\mathrm{dice}")=\frac{1}{6}+\frac{1}{6}=\frac{2}{6}\left(\frac{1}{3}\right)$               

• 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 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.