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

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   (∩ 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")
• 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 )

• •       “the probability of being less than four is 0.4”

How do I solve A level probability questions?

• Recall basic results of probability
  • 
    • It is important to understand that the above only applies if all outcomes are equally likely
  • 
    • 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!
• 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 

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

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