Statistics and Probability

Let's start with the single most important insight in every OC MR averages question: an average is just a sum in disguise. The moment you realise you can always recover the total by multiplying the average by the count, a whole family of questions becomes straightforward.

Mean and averages questions appear in almost every OC MR paper — usually one or two per test, spread across easy and difficult. The easy variants involve a single group; the harder ones split the group in two and ask you to work back and forth between both halves.

The examiner is checking whether you know that mean × count = total. Students who try to reason about averages directly — without converting to totals first — consistently pick the wrong answer on split-group questions.

You are told the average of a full group (e.g. 6 numbers averaging 18) and the average of one part of it (e.g. the first 3 average 14). You must find the average of the remaining part. The numbers are deliberately chosen so that trial-and-error is slow — the total-sum method is the only clean path.

25 marbles: 3 blue, 4 red, 8 yellow, 10 green. Which event is least likely? Count favourable outcomes for each: A=7, B=8, C=15, D=22, E=11. Smallest = 7 = Option A. The key skill: for 'not X' and 'neither X nor Y' events, subtract from the total.

Comparing probabilities by counting favourable outcomes is an easy-level staple in OC MR. The twist here — 'not X' and 'neither X nor Y' options — tests whether students can compute complementary event counts by subtracting from the total.

The examiner tests whether students can translate compound events ('either/or', 'not', 'neither/nor') into counts, then rank them. Option B (yellow, 8) is the classic distractor — students who stop at 'yellow is the rarest single colour' miss that blue+red combined (7) is actually fewer.

A bag or box contains coloured objects with given counts. One is drawn at random. Students evaluate several events (including compound 'not' and 'neither/nor' events) and identify the least or most likely.

Now let's tackle one of the most common probability formats in OC MR: a question that gives you a bag of coloured items and asks you to judge whether several probability statements are true or false. The key skill is checking each statement independently and knowing the complement rule cold.

Multi-statement probability questions appear in almost every OC MR paper from mock 10 onwards, usually in the medium-to-difficult range. They are a consistent score separator — students who check each claim systematically score them reliably; students who trust intuition frequently miss statement 3 (the complement).

The examiner is testing whether you can write a probability as a fraction (count of the event / total), compare fractions with the same denominator by comparing numerators, and correctly apply the complement rule: the chance of "not red" equals the chance of all other colours combined.

A bag contains coloured balls or marbles with different counts. One is drawn at random. Three probability claims are made — typically one comparing two colours, one claiming equality, and one involving "not [colour]". You must identify which subset of claims is correct.

Jess has counters numbered 1–11. Three statements are made about drawing one at random. Students must test ALL three carefully: Statement 1 fails because 1–11 has 6 odd but only 5 even numbers. Statement 2 fails because >6 and <6 each have 5 numbers (6 itself is excluded from both). Only statement 3 is true (each single counter has probability 1/11).

Multi-statement probability questions are a favourite at the medium tier of OC MR. They appear with counters, bags of balls, spinners, or dice. The key skill is testing every statement methodically rather than relying on intuition. Statement 1 here is the classic trap: most students assume odd and even are always balanced, forgetting that 1–11 is an odd count of numbers.

The examiner is testing three distinct probability concepts in one question: (1) recognising that an odd-sized set of consecutive integers has more odd than even numbers; (2) understanding that 'greater than 6' and 'less than 6' exclude 6 itself, making them equal; (3) knowing that all individual counters in a fair bag have equal probability. Students who get D (1 and 3) have fallen for the even/odd trap.

A set of numbered items is described. Three (or four) probability statements are given. Students must evaluate each statement independently and identify which subset is true. One statement is always a 'trap' that looks right at first glance (here: equal odd/even chance seems intuitive).

Shamira has 4×$5, 2×$10, and 2×$20 notes — 8 total. Three statements are given. Statement 1 (more likely $5 than $20) and Statement 2 (equally likely $5 or not-$5) are both true because exactly half the notes are $5. Statement 3 (certain to pick >$5) is false — there's a 50% chance of picking a $5 note.

Multi-statement probability questions involving a mixed collection (notes, balls, tiles) appear consistently in OC MR at the medium tier. Statement 2 here is particularly instructive: 'equally likely to take a $5 or not' is confirmed by the exact 4/8 = 1/2 split — an elegant application of complementary probability.

The examiner tests three probability concepts simultaneously: (1) comparing probabilities of different denominations, (2) recognising when an event and its complement are equally likely (probability = 1/2), (3) understanding 'certain' means probability = 1 (not just 'likely'). Statement 3 is the trap — half the notes are $5, so it is far from certain.

A collection of items is described by type and count. Three statements are made about drawing one at random. Students must evaluate all three using fractions: P(event) = favourable / total. A key trap is confusing 'likely' with 'certain'.

16 discs: 5R, 2G, 8B, 1Y. Statement 1 (certain blue) fails — P(blue)=½≠1. Statement 2 (equally likely blue or not blue) is true — P(blue)=8/16=½=P(not blue). Statement 3 (possible yellow) is true — P(yellow)=1/16>0. Answer: 2 and 3 only.

Evaluating probability language ('certain', 'equally likely', 'possible') with a statement-combination format is a medium-level OC MR staple. Students must know the precise threshold for each term: certain=1, equally likely=equal fractions, possible=>0.

The examiner tests precise probability vocabulary. Statement 1 ('certain') traps students who see 8 blue discs and think 'there are more blue than any other colour' without realising certain means probability = 1. Statement 3 ('possible') checks if students understand that even 1 disc out of 16 is a possible outcome.

A set of coloured objects is described with counts. Three statements each use a probability term ('certain', 'equally likely', 'possible', 'impossible', 'more likely', etc.). Students evaluate each statement and identify which are correct.