Binomial and geometric questions look routine, and the method usually is. The reports show that the marks go missing when the wording is turned into a probability statement, and when the number of arrangements is forgotten. 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
- Misreading the success criteria. "More than 6 throws" interpreted as including throw 6; including 8 in a group where the question did not; P(X ≤ 10) or P(X = 9) in place of the required condition; and a very common error of P(X ≤ 7) − P(X > 3). Reports say students should understand the vocabulary used in these contexts.
- Subtracting cumulative probabilities. In P(X ≤ 7) − P(X ≤ 2), leaving out P(3) or forgetting the "1 −" for a probability of the form P(X ≥ ...). In P(X ≤ 6) − P(X ≤ 1), incorrectly subtracting P(X ≤ 2).
- Complement errors. After adding the probabilities of 7, 8 and 9, forgetting to subtract from 1, or leaving out 7. With the longer method, leaving out the case of none.
- Missing coefficients. Forgetting "× 7C1", forgetting to multiply by 4 (giving 0.0181), using 5C2 when the last trial had to be a success, or failing to see that a first success could occur in 9C1 ways. The reports describe these as common misconceptions.
- Missing terms. Omitting the third term for two people supporting a team, or not including the probability of the second 6.
- Geometric versus binomial. In one question some candidates confused the situation with a geometric distribution. In another, they treated "at least 2 heads for the first time on the seventh throw" as "exactly 2" or "exactly 3" heads, and wrongly assumed that adding separate probabilities for the success criteria is equivalent to the correct calculation.
- Rounding. Rounding the probability of each term, so that the final answer was outside the acceptable range (in one report the most accurate value seen, often 0.09955, was outside the acceptable range). Others confused three decimal places with three significant figures and dropped the last digit.
- Listing outcomes. Where students listed all possibilities and added them, they were often not systematic, and missed some.
- Using a Normal approximation. In one question a few candidates tried to use a Normal approximation to the binomial.
- Mean and variance for the approximation. Not writing the unsimplified calculations for the mean and variance, confusing variance with standard deviation, and no continuity correction (or the wrong one) when approximating a discrete distribution.
What examiners advise
- Translate the words into a statement about X, such as P(X ≥ 7) = 1 − P(X ≤ 6), before touching the calculator. Check the end values.
- A simple diagram or list helps you to interpret the success criteria.
- Check what is being multiplied: the probability of each outcome and the number of ways it can occur.
- Do not round individual terms. Keep full values until the end, and evaluate the entire expression on the calculator.
- For a Normal approximation, state the unsimplified mean and variance, and think carefully about the continuity correction in the light of the success criteria.
Habits that help
- Ask which of "binomial" or "geometric" matches the question by looking at whether there is a fixed number of trials.
- For "at least" and "at most", write out the first and last values you want.
- Check that your answer is a probability between 0 and 1.
Questions students ask
How do I decide between a binomial and a geometric distribution?
Look at whether there is a fixed number of trials. A geometric question asks for the trial on which the first success occurs. In one question students treated "at least 2 heads for the first time on the seventh throw" as "exactly".
How do I translate "more than", "at least" and "at most" into probabilities?
Write the statement about X before touching the calculator, for example P(X ≥ 7) = 1 − P(X ≤ 6), and check the end values. Write out the first and last values you want.
Why does my cumulative subtraction go wrong?
In P(X ≤ 7) − P(X ≤ 2), students leave out P(3) or forget the "1 −" in P(X ≥ …). Check what is being subtracted.
Which coefficients do students forget?
The number of ways. Examples are forgetting × 7C1 or a multiplication by 4, using 5C2 when the last trial had to be a success, and omitting a term. Check what is being multiplied: the probability of each outcome and the number of ways it can occur.
Should I round each term before adding?
No. Rounding the probability of each term put one answer outside the acceptable range. Keep full values and evaluate the whole expression on the calculator.
What do I need for a Normal approximation to the binomial?
State the unsimplified mean and variance, do not confuse variance with standard deviation, and think about the continuity correction in the light of the success criteria.
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.