Further Probability (Extended)

Bringing It All Together

Further Probability is the most demanding topic in Unit 6 at Extended level. It combines the skills developed across all previous probability topics — including conditional probability, tree diagrams, and counting methods — into multi-step problems that require careful reasoning.

Multi-Step Tree Diagrams with Conditional Probability

Tree diagrams become more complex when probabilities on later branches depend on what happened at an earlier branch. This is the key feature that distinguishes further probability from the basic trees encountered at Standard level.

For these problems:

• Assign conditional probabilities to each branch based on the outcome of the previous stage
• Multiply along branches to find the probability of a complete sequence
• Add the probabilities of all paths that satisfy the required condition

Mutually Exclusive vs Independent Events

These two concepts are often confused but are fundamentally different:

• Mutually exclusive events cannot both occur at the same time. If A and B are mutually exclusive, P(A and B) = 0, and P(A or B) = P(A) + P(B).
• Independent events do not influence each other. If A and B are independent, P(A and B) = P(A) × P(B), but they can both occur.

Note: If two events are mutually exclusive and both have non-zero probability, they cannot be independent.

Combined Problem Types

Expect questions that require you to first decide which probability rule or tool to apply, then execute multi-step calculations. Common formats include:

• Tree diagrams where the second stage involves conditional probabilities
• Problems combining two-way table reading with formal probability rules
• Identifying whether events are independent, mutually exclusive, or neither — and justifying your answer with probability values

Common Mistakes

• Treating mutually exclusive events as independent (they are almost never both simultaneously)
• Forgetting to consider all paths in a tree diagram when summing probabilities
• Applying the simple multiplication rule to dependent events

Criterion B and D in Further Probability

Criterion B tasks may ask you to investigate whether two events described in a dataset are independent. Criterion D tasks often embed these calculations in contexts — reliability of systems, genetics, or financial risk — where interpreting your final answer matters as much as calculating it correctly.