4 high-yield, syllabus-aligned questions on Statistics & probability, covering 14 individually creditable mark points. Each one shows the answer that scores full marks under our guide, which phrase earns which mark, and a common incomplete answer — so you can see the difference rather than guess at it.
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Every question and solution is reviewed by examiners versed in WAEC and JAMB, each with more than thirty years of experience.
Mark allocations are TopMarks’ own, written against the published syllabus and Chief Examiners’ reports. They are a guide to how these answers are usually credited, not an official marking scheme.
Calculate4 marks
The marks scored by five students are 4, 6, 8, 10 and 12. Calculate the mean and the standard deviation of the marks.
TopMarks model answer
Mean = (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8. The deviations from the mean are -4, -2, 0, 2 and 4, and their squares are 16, 4, 0, 4 and 16. The sum of the squared deviations = 40. Variance = 40 ÷ 5 = 8. Standard deviation = √8 = 2.83.
Suggested marking guide
1Correct mean
2Squared deviations found
3Variance calculated
4Correct standard deviation
The working, line by line
1Mean = 40 ÷ 5 = 8 — Everything else depends on this, so calculate it first and check it.
2Deviations: -4, -2, 0, 2, 4 — Subtract the mean from each value. The deviations must sum to zero — a free check on your mean.
3Squared deviations: 16, 4, 0, 4, 16; sum = 40 — Squaring removes the minus signs, which is why it is done.
4Variance = 40 ÷ 5 = 8; standard deviation = √8 = 2.83 — The standard deviation is the SQUARE ROOT of the variance. Stopping at 8 answers a different question.
Why it scores
The deviations must add up to zero. That is a free check on the mean before you square anything, and it catches the arithmetic slip that would otherwise poison every later step. Note also the last line: variance and standard deviation are not the same, and the question asked for the standard deviation.
A common incomplete answer
“Mean = 8”
What it costs: Three marks of four. The mean is correct and credited, but the standard deviation — the larger part of the question — is not attempted.
Key ideas to include: mean, deviation from the mean, squared deviations, variance, standard deviation.
State3 marks
State the difference between the mean, the median and the mode.
TopMarks model answer
The mean is the sum of all the values divided by the number of values. The median is the middle value when the data are arranged in order of size, or the average of the two middle values when the number of values is even. The mode is the value that occurs most frequently in the data.
Suggested marking guide
1Mean defined
2Median defined, including arranging in order
3Mode defined
Why it scores
The median definition must include ARRANGING IN ORDER, because that is the step candidates forget when they use it. Picking the physically middle number of an unsorted list is the commonest median error in the whole paper, and the definition is where the habit is formed.
A common incomplete answer
“The mean is the average, the median is the middle one and the mode is the most common.”
What it costs: One mark of three. The mode is credited. “The average” does not define the mean, and the median is given without the essential step of arranging the data in order of size.
Key ideas to include: mean, median, mode, arranged in order, frequency.
Find3 marks
A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random one after the other without replacement. Find the probability that both are red.
TopMarks model answer
There are 10 balls in all. P(first is red) = 4/10. After a red ball has been removed and not replaced, 3 red balls remain out of 9. So P(second is red, given the first was red) = 3/9. Therefore P(both red) = 4/10 x 3/9 = 12/90 = 2/15.
Suggested marking guide
1Correct first probability
2Second probability reflects the ball not being replaced
3Correct final probability
The working, line by line
1Total balls = 4 + 6 = 10; P(first red) = 4/10 — Count the total first. It is where the denominator comes from.
2WITHOUT replacement: one red has gone, so 3 reds remain out of 9 balls — BOTH numbers change — the reds fall from 4 to 3 AND the total falls from 10 to 9. Changing only one is the classic half-error.
3P(both red) = 4/10 x 3/9 = 12/90 = 2/15 — "Both" means AND, and AND means multiply.
Why it scores
The words “without replacement” are the entire question. They change both the numerator and the denominator for the second draw. Circle those words when you see them, and write the second fraction only after you have asked what is left in the bag.
A common incomplete answer
“P = 4/10 x 4/10 = 16/100”
What it costs: Two marks of three. The first probability is credited, but the ball was not replaced, so the second draw is from 9 balls containing 3 reds, not from the original 10.
Key ideas to include: without replacement, conditional probability, multiply for AND, sample space.
Explain4 marks
Explain what is meant by mutually exclusive events and by independent events, giving one example of each, and state how probabilities are combined in each case.
TopMarks model answer
Two events are mutually exclusive if they cannot both occur at the same time, so the occurrence of one excludes the other; obtaining a head and obtaining a tail on a single toss of a coin are mutually exclusive. For mutually exclusive events the probabilities are added. Two events are independent if the occurrence of one does not affect the probability of the other; the outcomes of two separate tosses of a coin are independent. For independent events the probabilities are multiplied.
Suggested marking guide
1Mutually exclusive defined
2Probabilities are added for mutually exclusive events
3Independent defined
4Probabilities are multiplied for independent events
Why it scores
The rule to carry into the exam is short: OR means add, AND means multiply. Almost every probability question in the paper reduces to deciding which of those two words the situation means, and the two definitions here are what let you decide.
A common incomplete answer
“Mutually exclusive events cannot happen together, and independent events do not affect each other.”
What it costs: Two marks of four. Both definitions are correct and credited, but neither example is given and neither combination rule — add for OR, multiply for AND — is stated.
Practise Statistics & probability on real questions
Reading a full-mark answer is the first half. Writing one under time is the other. A free account opens exam-standard practice in Mathematics with the full solution on every question.