Discrete random variable questions ask for a probability distribution table, followed by E(X) and Var(X). The reports show that students know the formulae, and the marks go missing in the table: outcomes left out, probabilities that do not total 1, and a lack of working. 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
- Missing outcomes. The most common error is to leave out a value of X, usually X = 0 (no heads, no yellow marbles, no red balls). Because of this the probabilities did not sum to 1.
- Impossible outcomes. Listing values that could not be achieved, such as 4 from the values on the spinners, or $10 as a possible outcome, often from an outcome table made earlier in the question.
- Missing scenarios. Finding P(X = 2) from one scenario and leaving out the others. Reports also note that students who used an outcome table assumed that the number of coins of each value, or the order, did not matter.
- Missing the number of ways. Multiplying the correct probabilities and forgetting the number of ways they can occur (the combination term such as 6C2).
- Probabilities that do not total 1, including totals greater than 1. The reports describe a total that is not 1 as a prompt that an error has been made.
- Variance. Not squaring the mean within the variance formula (a common error in several reports), omitting to subtract the mean, or squaring the mean before subtracting.
- Misreading the definition of X. In one question, students ignored the last four words of the definition and used "the larger score" instead of "the larger score minus the smaller score".
- Unsupported values. Stating E(X) without working. Reports say this earns no credit, especially when it comes from an incorrect table. In a "show that" question, a true statement such as 5/8 × 3/7 = 15/56 is not a justification.
- A poor table. Candidates did not present a clear table, and did not use lines to give it structure.
- Exact answers. Giving 0.938 as the final answer without stating the exact value (15/16 or 0.9375).
- Negative values. Weaker solutions struggled with negative values of X, with arithmetical errors.
What examiners advise
- List all the possible values of X first, including 0 where it is possible, and check each one can really happen.
- Check that the probabilities total 1 before you go on.
- Show the unevaluated expression for E(X) and Var(X). Reports say this lets you earn the method mark even when the table is wrong.
- Show where each value comes from, particularly in a "show that" question.
- Use a ruler for the table, and a common denominator for the probabilities if that helps you to see what is going on.
- Fractions need not be decimals.
Habits that help
- Write each outcome next to the scenarios that produce it, then count the scenarios.
- Compute E(X²) in a separate line so that you can see the squaring.
- After finding Var(X), check that it is positive.
Questions students ask
What do I put in the probability distribution table?
List all the possible values of X first, including 0 where it is possible, and check each can really happen. Leaving out X = 0 is the most common error, and it stops the probabilities summing to 1.
Why must the probabilities total 1?
It is a check. A total that is not 1, or is greater than 1, is a prompt that an error has been made. Check before you go on.
How do I find the variance of a discrete random variable?
Compute E(X²) on a separate line and subtract the square of the mean: Var(X) = E(X²) − [E(X)]². Students forgot to square the mean, omitted to subtract it, or squared it before subtracting. Show the unevaluated expression so you can earn the method mark.
What do I do in a "show that" question about a probability?
Show where each value comes from. A true statement such as 5/8 × 3/7 = 15/56 is not a justification on its own.
Should I give exact answers or decimals?
Exact. Giving 0.938 without stating the exact value, 15/16 or 0.9375, lost the mark. Fractions need not be decimals.
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.