M: Probability: Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables. Understand and use the conditional probability formula. P A B = P A ∩B P B

The idea

Conditional probability is the probability of an event given that another event has already occurred. You must understand and use it with tree diagrams, Venn diagrams and two-way tables, and apply the formula P(A|B) = P(A ∩ B) / P(B). For example, if a bag has 3 red and 2 blue counters, the probability of a second red given a first red is 2/4. Tree diagrams show this by changing probabilities on the second set of branches. Venn diagrams show it as the proportion of B that lies inside A. Two-way tables give counts, so you divide the count in the overlap by the total for the condition. The formula works when P(B) > 0. Always identify the condition and restrict the sample space to that event.

What earns the marks

• Correctly identify the condition event and restrict the sample space to that event when calculating a conditional probability.
• Use the formula P(A|B) = P(A ∩ B) / P(B) accurately, ensuring that P(B) is not zero.
• Interpret and construct tree diagrams where probabilities on the second set of branches are conditional on the first outcome.
• Use Venn diagrams to find conditional probabilities by dividing the probability of the intersection by the probability of the condition.
• Extract conditional probabilities from two-way tables by dividing the relevant cell count by the row or column total for the condition.
• Recognise and distinguish between P(A|B) and P(B|A), and between conditional and unconditional probabilities.

Common mistakes and how to put them right

• Confusing P(A|B) with P(B|A).

  Put it right: Check which event is given; the condition is the event that has already happened.

• Using the wrong denominator in the conditional probability formula, such as dividing by P(A) instead of P(B).

  Put it right: The denominator is always the probability of the condition.

• Forgetting to update probabilities on the second set of branches of a tree diagram when sampling without replacement.

  Put it right: Adjust the numerator and denominator to reflect the reduced sample space.

• Misreading a two-way table by using the grand total instead of the row or column total for the condition.

  Put it right: Identify the condition and use the corresponding marginal total.

Exam tips

• Always write down the formula P(A|B) = P(A ∩ B) / P(B) before substituting values; this makes your method clear and can earn method marks.
• When using a tree diagram, label each branch with its probability and remember that second-level probabilities are conditional on the first outcome.
• For two-way tables, annotate the table with row and column totals to help you pick the correct denominator for the condition.
• Check whether events are independent: if P(A|B) = P(A), then A and B are independent, which can simplify calculations.