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