IB DP Maths: AI SL复习笔记4.3.3 Sample Space Diagrams

What is a Venn diagram?

• A Venn diagram is a way to illustrate events from an experiment and are particularly useful when there is an overlap between possible outcomes
• A Venn diagram consists of
  • a rectangle representing the sample space (U)
    • The rectangle is labelled U 
    • Some mathematicians instead use S or ξ 
  • a circle for each event
    • Circles may or may not overlap depending on which outcomes are shared between events
• The numbers in the circles represent either the frequency of that event or the probability of that event
  • If the frequencies are used then they should add up to the total frequency
  • If the probabilities are used then they should add up to 1

What do the different regions mean on a Venn diagram?

• Venn diagrams show ‘AND’ and ‘OR’ statements easily
• Venn diagrams also instantly show mutually exclusive events as these circles will not overlap
• Independent events can not be instantly seen
  • You need to use probabilities to deduce if two events are independent

How do I solve probability problems involving Venn diagrams?

• Draw, or add to a given Venn diagram, filling in as many values as possible from the information provided in the question
• It is usually helpful to work from the centre outwards
  • Fill in intersections (overlaps) first
• • Add together the frequencies/probabilities in the B circle
    • This is your denominator
  • Out of those frequencies/probabilities add together the ones that are also in the A circle
    • This is your numerator
  • Evaluate the fraction

What is a tree diagram?

• A tree diagram is another way to show the outcomes of combined events
  • They are very useful for intersections of events
• The events on the branches must be mutually exclusive
  • Usually they are an event and its complement
• The probabilities on the second sets of branches can depend on the outcome of the first event
  • These are conditional probabilities
• When selecting the items from a bag:
  • The second set of branches will be the same as the first if the items are replaced
  • The second set of branches will be the different to the first if the items are not replaced

How are probabilities calculated using a tree diagram?

• To find the probability that two events happen together you multiply the corresponding probabilities on their branches
  • It is helpful to find the probability of all combined outcomes once you have drawn the tree
• To find the probability of an event you can:
  • add together the probabilities of the combined outcomes that are part of that event
    • Do I have to use a tree diagram?

• If there are multiple events or trials then a tree diagram can get big
• You can break down the problem by using the words AND/OR/NOT to help you find probabilities without a tree
• You can speed up the process by only drawing parts of the tree that you are interested in

Which events do I put on the first branch?

• If the events A and B are independent then the order does not matter
• If the events A and B are not independent then the order does matter
  • If you have the probability of A given B then put B on the first set of branches
  • If you have the probability of B given A then put A on the first set of branches