Probability questions in Statistics 1 are about tree diagrams, conditional probability, independence and the number of ways an event can happen. The reports show that the same slips happen year after year: replacement, missing scenarios and a missing multiplier. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Assuming replacement. One of the most common errors in the reports. A question said eggs were eaten, or a marble was not replaced, and students used the same probabilities on the second pick. In another question, students used probabilities that assumed the discs were replaced.
- Tree diagram faults. Branches left off (only one of the Pass branches extended), branches continued after a failure, branches not labelled fully (the words Pass and Fail were the most common omission), and a missing branch with a non-zero probability, such as the Green branch for Bag B.
- Labelling the first set of branches with the names of the sets (X and Y) instead of the outcomes (H and T).
- A diagram that does not match the situation. In one question, students did not realise that once the bell had rung, Amy no longer played, so their trees were wrong. In another, a third sweet was added to the tree that the question did not ask for.
- Missing scenarios. Leaving out one of the ways to get the result, such as P(TBR) or P(TRR), or P(Red, Red) in the success criteria. An inconsistent way of listing outcomes made this worse.
- Missing the number of orders. Forgetting to multiply by 6, or by 3, when the objects could be chosen in different orders. Another report says that the number of arrangements for WBB is not the same as for WWB, because of the repeated colour.
- Exact values rounded. Giving a 3 significant figure answer when an exact value had been calculated. Reports say that only non-exact answers should be rounded.
- Conditional probability. Calculating only the probability of both events and not dividing, using a 3 significant figure value in the denominator (4 significant figures or more are needed to justify a 3 significant figure answer), or forgetting to include given information in the denominator, as in "when Suki has milk, she never has a biscuit".
- Independence. Assuming independence and finding P(S ∩ T) as P(S) × P(T). In another session, students had no strategy for deciding about independence, used only words, or did not put the probabilities in a comparable form.
- Using the wrong denominator carried forward from an earlier part.
- Misreading numbers. In one question 177 was read as 117, and both values were used in the same calculation.
- Probabilities greater than 1. These appear in many reports, and each time the reports say they should have prompted the student to check.
What examiners advise
- Draw a tree diagram or table. Reports say a visual representation often leads to an efficient solution, and that it clarifies the relationships you need to use.
- Use a consistent method of listing outcomes so that nothing is missed.
- Show the formula for conditional probability and the values you substitute. Workings must communicate your plan even if the final answer is correct.
- Put the probabilities in a comparable form (all fractions, or all decimals) before testing independence, and have a clear strategy for the test.
- Evaluate the whole unsimplified expression on the calculator to avoid rounding and re-keying errors.
- Leave probabilities as fractions when they are exact. There is no expectation to state a decimal.
Habits that help
- Before you start, underline words such as "replaced", "eaten", "not replaced" and "given that".
- After you list the scenarios, count them and ask yourself whether each could happen in more than one order.
- Add up all of your final probabilities when the question has a complete set of outcomes. They should come to 1.
Questions students ask
How do I know whether to use replacement in a probability question?
Underline words such as "replaced", "eaten", "not replaced" and "given that" before you start. Assuming replacement when the item was not replaced is one of the most common errors in the reports.
How should I draw a tree diagram?
Label the branches fully with the outcomes (such as H and T, or Pass and Fail), not with the names of sets. Do not continue branches after a failure when the situation stops, and make sure the diagram matches the situation.
How do I use the conditional probability formula?
Show the formula and the values you substitute. Students found only the probability of both events and did not divide, or used a 3 significant figure value in the denominator. Use 4 significant figures or more, or keep it exact.
How do I test whether two events are independent?
Put the probabilities in a comparable form, all fractions or all decimals, and compare P(S ∩ T) with P(S) × P(T). Do not assume independence, and do not just argue in words.
How do I make sure I have all the scenarios?
Use a consistent method of listing outcomes. Students left out ways to get the result, forgot to multiply by the number of orders, and sometimes included impossible cases. If you have a complete set of outcomes, the probabilities should add to 1.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.