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WAEC 2008 · Paper 2 · Q10

In a class of 200 students, 70 offered Physics, 90 Chemistry and 100 Mathematics, while 24 did not offer any of the three subjects. Twenty-three students offered Physics and Chemistry, 41 Chemistry and Mathematics, while 8 offered all three subjects.

  1. (a)

    Draw a Venn diagram to illustrate the information.

    Model answer
    P (70)C (90)M (100)273439153320824U

    Let xx be the number who offered Physics and Mathematics only. The regions are: all three 8; P and C only 15; C and M only 33; P and M only xx; P only 47−x47 - x; C only 34; M only 59−x59 - x; none 24. They add up to 200, so x=20x = 20.

  2. (b)

    Find the probability that a student selected at random from the class offered: (i) Physics only; (ii) exactly two of the subjects.

    Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

  1. Start in the middle: 8 offered all three subjects.
  2. Physics and Chemistry only: 23−8=1523 - 8 = 15.
  3. Chemistry and Mathematics only: 41−8=3341 - 8 = 33.
  4. Let xx offer Physics and Mathematics only.
  5. Then Physics only is 70−15−8−x=47−x70 - 15 - 8 - x = 47 - x, Chemistry only is 90−15−8−33=3490 - 15 - 8 - 33 = 34 and Mathematics only is 100−33−8−x=59−x100 - 33 - 8 - x = 59 - x.
  6. All the regions, with the 24 outside, add up to 200: (47−x)+34+(59−x)+15+33+x+8+24=200(47 - x) + 34 + (59 - x) + 15 + 33 + x + 8 + 24 = 200, so 220−x=200220 - x = 200 and x=20x = 20.
  7. So Physics only is 2727, Mathematics only is 3939 and Physics and Mathematics only is 2020.
  8. Fill these into the diagram.

(b)(i)

  1. P(Physics only)=27200P(\text{Physics only}) = \frac{27}{200}.

(ii)

  1. Exactly two subjects: 15+33+20=6815 + 33 + 20 = 68 students, so the probability is 68200=1750\frac{68}{200} = \frac{17}{50}.

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