Probability-Tree-Diagrams Wjec-Eduqas Higher

Tree diagrams

How do I draw a tree diagram?

• Tree diagrams can be used for repeated experiments with two outcomes

  • The 1st experiment has outcome A or not A

  • The 2nd experiment has outcome B or not B

• Read the tree diagram from left to right along its branches 

  • For example, the top branches give A followed by B

    • This is called A and B

How do I find probabilities from tree diagrams?

• Write the probabilities on each branch

  • Remember that P(not A) = 1 – P(A)

    • Probabilities on each pair of branches add to 1

• Multiply along the branches from left to right

  • This gives P(1st outcome and 2nd outcome)

• Add between the separate cases

  • For example

    • P(AA or BB) = P(AA) + P(BB)

• The probabilities of all possible cases add to 1

• If asked to find the probability of at least one outcome, it is quicker to do 1 – P(none)

How do I use tree diagrams with conditional probability?

• Probabilities that depend on a particular thing having happened first in a tree diagram are called conditional probabilities

• For example, the probability that a team wins a game may depend on whether they won or lost the previous game

  • The probabilities for ‘win’ on the first set of branches may be different to those for ‘win’ on the second set of branches

• Another example of conditional probabilities is “without replacement” scenarios

  • e.g. two items are drawn from a bag of different coloured items without the first item drawn being replaced

  • The probabilities on the second set of branches will change depending on which branch has been followed on the first set of branches

    • The denominators in the probabilities for the second set of branches will be one less than those on the first set of branches

    • The numerators on the second set of branches will also change

• Conditional probability questions are sometimes introduced by the expression ‘given that…’

  • e.g. ‘Find the probability that the team win their next game given that they lost their previous game’

• The notation is often used for conditional probabilities

  • That is read as ‘the probability of A given B’

  • e.g.  is the probability a team wins, given that they lost the previous game