EPFO EO/AO 2023 General Ability Test · Question 40 of 120

Correct — C, (c) 21. Start with the phrase the booklet emphasises. 'Not less than' means greater than OR EQUAL TO — the relation ≥, not the strict inequality >. The bold italic on 'not' is there to stop the eye sliding over it, because the whole difference between the right answer and option (a) is whether the cases where the two throws are equal are counted. Note that this is a relation inside the condition, not a question asking which option is false. Now count. Two throws of a die give 6 × 6 = 36 ordered outcomes. Split them into three groups: • first equals second: (1,1) up to (6,6) — 6 outcomes. • first greater than second, and first less than second: these two groups have the same size, since swapping the two throws matches each outcome of one with exactly one outcome of the other. Together they account for 36 − 6 = 30 outcomes, so each has 15. The condition 'first ≥ second' covers the equal group and the greater group: 6 + 15 = 21. The direct count confirms it. If the second throw is 1, the first may be any of 1 to 6 — 6 ways; if the second is 2, the first may be 2 to 6 — 5 ways; and so on down to a second throw of 6, which allows only 6 on the first — 1 way. That is 6 + 5 + 4 + 3 + 2 + 1 = 21. One more check on what is being asked: the stem says 'the number of ways', so the answer is a count, not a probability. The probability would be 21/36 = 7/12, which is not among the options — a useful confirmation that the count is what was wanted. (The booklet writes 'A dice is thrown', using 'dice' where the singular is 'die'; the stem is reproduced as printed.)

• (a)15 — This is the count for 'first strictly GREATER than second', and it is the designed trap of the item — the answer you get by reading 'not less than' as 'more than' and dropping the six outcomes where the two throws match. The emphasis the booklet puts on 'not' is aimed exactly here. Fifteen is also C(6,2), the number of ways to choose two different faces, which makes it feel like a properly derived answer.
• (b)20 — Twenty corresponds to no count in this problem. It is one short of the right answer and five more than the other real figure, sitting between them as a plausible round number. There is no way to reach it by adding or omitting a natural group of outcomes, which is worth noticing: in counting problems the wrong answers that matter are the ones with a derivation, and this one has none.
• (d)36 — Thirty-six is the total number of ordered outcomes when a die is thrown twice, that is, the answer if the condition is ignored altogether. It is the correct answer to the sample space, and it is a useful number to have written down — but as the answer to this stem it means the constraint was never applied.

Throwing a die twice produces an ordered pair, and the sample space of 36 equally likely outcomes is the foundation of nearly every dice question. The technique this item rewards is symmetry counting: the outcomes where the first exceeds the second and those where the second exceeds the first are in one-to-one correspondence through swapping, so each group is half of whatever remains after the ties are removed. That single observation replaces a case-by-case enumeration. The equivalent formula is that the number of ordered pairs with first ≥ second is C(6,2) + 6 = 15 + 6 = 21, or equivalently C(7,2) = 21, since choosing a pair of values with repetition allowed and then assigning the larger to the first throw is the same as choosing 2 items from 6 with repetition.

Probability and counting items in this paper are deliberately small — one or two dice, a pack of cards, a handful of balls — and the difficulty is loaded into the wording rather than the mathematics. 'Not less than', 'at most', 'at least' and 'no more than' are the phrases that decide these questions, and the Commission underlines the crucial word rather than hiding it. The habit rewarded is translating the condition into a symbol (≥, ≤, >, <) before counting anything, so that the boundary cases are settled in advance.

• 'Not less than' means ≥; 'not more than' means ≤; both include the boundary case.
• Two throws of a die give 6 × 6 = 36 ordered outcomes.
• Outcomes with the two throws equal: 6.
• Outcomes with first greater than second: (36 − 6) ÷ 2 = 15, by the swap symmetry.
• Outcomes with first ≥ second: 15 + 6 = 21.
• Direct count: 6 + 5 + 4 + 3 + 2 + 1 = 21.
• As a formula, C(6,2) + 6 = C(7,2) = 21.
• The corresponding probability would be 21/36 = 7/12, but the stem asks for a number of ways, not a probability.
• The singular of 'dice' is 'die'; the booklet uses 'dice' for one.

• Reading 'not less than' as 'greater than' and losing the six equal-value outcomes.
• Answering with the probability when the stem asks for the number of ways, or the reverse.
• Treating the two throws as unordered, which halves the sample space and breaks the count.
• Forgetting that the swap symmetry applies only after the ties have been set aside — the equal outcomes map to themselves.

EPFO EO/AO keeps its probability and counting items elementary: dice, coins, cards, and simple selections. What varies is the wording of the condition. Rehearsing the inequality translations is worth more preparation time than the counting itself, because the arithmetic in these questions is never the obstacle.

No directly related past PYQ was found.

• practice — not a real PYQ

  A die is thrown two times. In how many ways can the number on the first throw be greater than the number on the second throw ?

  • (a)6
  • (b)15
  • (c)21
  • (d)30

  Answer(b) 15

• practice — not a real PYQ

  Two dice are thrown together. What is the probability that the sum of the numbers appearing on them is 9 ?

  • (a)1/6
  • (b)1/9
  • (c)1/12
  • (d)5/36

  Answer(b) 1/9