TTMUA Lab
TMUA 2021 · Paper 2 · Question 3 of 20

TMUA 2021 Paper 2 Question 3

Logic and arguments — Necessary conditions · probability from a class. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Logic and argumentsNecessary conditions · probability from a class8 options
A student is chosen at random from a class. Each student is equally likely to be chosen.

Which of the following conditions is/are necessary for the probability that the student wears glasses to equal 415?

I    Exactly 11 students in the class do not wear glasses.
II   The number of students in the class is divisible by 3.
III The class contains 30 students, and 8 of them wear glasses.

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · C
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
A condition is necessary only if it holds in every case that gives probability 415. Since 415 is in lowest terms, the class must have 15k students of whom 4k wear glasses, for some positive integer k. Then 15k is always a multiple of 3, so II is necessary. But k = 2 gives 30 students, 8 with glasses and 22 without, which kills I; and k = 1 gives a class of 15, which kills III.