Probability Tree Diagrams | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Tree Diagrams

What is a tree diagram?

• A tree diagram is utilized for:
  • Displaying outcomes of sequential events
  • Assisting in probability calculations involving AND and OR scenarios
• Tree diagrams are predominantly employed when an event has only two significant outcomes such as:
  • "Rolling a 6 on a dice"
  • "Not rolling a 6 on a dice"
• These outcomes are mutually exclusive, meaning they cannot occur simultaneously.

How do I draw and label a tree diagram?

• The initial branches depict the results of the first experiment, which can be labeled as "A" and "not A".
• Two distinct sets of branches illustrate the outcomes of the second experiment:
  • The first set continues from "A" in the first experiment.
  • The second set continues from "not A" in the first experiment.
• The outcomes of the second experiment are broadly termed as "B" and "not B".
• Probabilities for each specific outcome are indicated along the branches of the tree diagram.
• At the conclusion of the diagram, various combinations of the two experiments are compiled, including:
  • "A" and "B"
  • "A and not B"
  • "not A and B"
  • "not A and not B"

How do I solve probability problems involving tree diagrams?

• Interpret questions in terms of AND and/or OR
• Draw, or complete a given, tree diagram
  • Determine any missing probabilities
    • often using 1 - P (A)
• Write down the outcomes of both events and work out their probabilities
  • These are AND statements
  • 
  • You may see this as “Multiply along branches”
• If more than one outcome is required then add their probabilities
  • These are OR statements
  • 
  • You may see this as “Add different outcomes”
• When you are confident with tree diagrams you can just pull out the outcome(s) you need
  • you do not routinely have to work all of them out

How do I use tree diagrams with conditional probability?

• Conditional Probabilities in Tree Diagrams:
  • Probabilities that depend on a prior event are called conditional probabilities.
  • Example: A team's probability of winning or losing a game may change based on the outcome of their previous game.
• Identifying Conditional Probabilities in Tree Diagrams:
  • Interested in the probability of winning after losing the previous game.
  • This probability is found in the branches following the 'lose' outcome in the first set of branches.
• Drawing Tree Diagrams for Sequential Events:
  • Example: Drawing two counters from a bag without replacement.
  • Probabilities on the second set of branches depend on the outcome of the first draw.
• Adjusting Probabilities on Second Set of Branches:
  • Denominators for the second set of probabilities decrease by one compared to the first set.
  • Numerators also change based on the first event's outcome.
• Using Worked Examples:
  • Look at worked examples to understand how conditional probabilities are calculated and represented.
• Identifying Conditional Probability Questions:
  • Often introduced with "given that...".
  • Example: "Find the probability that the team wins their next game given that they lost their previous game."
• Straight Bar Notation:
  • Conditional probabilities are written using the 'straight bar' notation P(A∣B).
  • Read as "the probability of A given B".
  • Example: P(win∣lose) means the probability of winning given that they lost the previous game.
• Order of Events in Straight Bar Notation:
  • The event after the straight bar occurs first.
  • The event before the straight bar occurs afterward.

Combined Probability

What is Meant by Combined Probabilities?

• In general, combined probabilities involve considering more than one event. 
• These events can be independent or mutually exclusive. 
• They may also include events that follow on from previous events, such as rolling a dice followed by flipping a coin.

How do I work with and calculate combined probabilities?

• AND/OR Statements:
  • Convert questions into probability statements using AND (for combined events) and OR (for alternative events).
  • Example: "The probability of rolling a 6 followed by flipping heads" becomes "the probability of rolling a 6 AND flipping heads."
• Using AND and OR in Probability:
  • AND means multiply (used for independent events).
  • OR means add (used for mutually exclusive events).
• Total Probability Equals 1:
  • The sum of all probabilities in a given scenario is 1.
  • This is particularly useful for events happening or not happening.
  • 
• Tree Diagrams for Combined Probabilities:
  • Tree diagrams help calculate combined probabilities, especially when considering two outcomes from each event.
  • Example: The probability of being stopped at one set of traffic lights and also at a second set.
• Using Rules Without Diagrams:
  • Unless specifically required, drawing a diagram is not necessary.
  • Often quicker to consider possible outcomes and apply AND/OR rules directly without a diagram.