WAEC 2022 · Paper 2 · Q9

In a class of 50 students, 18 play football, 20 volleyball and 21 handball. 9 play football and volleyball, 6 football and handball, 11 volleyball and handball, while 15 play none of the 3 games.

(a)
Represent the information in a Venn diagram.
Model answer
Let $x$ play all three. Then football only is $18 - (9 - x) - (6 - x) - x = 3 + x$ , volleyball only is $20 - (9 - x) - (11 - x) - x = x$ and handball only is $21 - (6 - x) - (11 - x) - x = 4 + x$ . The regions add up to $50 - 15 = 35$ , which gives $x = 2$ : the regions are then $5, 2, 6, 7, 4, 9$ and 2.
(b)
How many students play: (i) all the three games? (ii) only one game? (iii) only football?
Separate values with commas, e.g. 3, −2
(c)
If a student is chosen at random from the class, find the probability that the student played only two games.

Worked solution (try it first)

(a)

Draw three overlapping circles F, V and H in a rectangle of 50 students, with 15 outside all the circles.
Let $x$ play all three.
The 35 who play at least one game satisfy $18 + 20 + 21 - 9 - 6 - 11 + x = 35$ , so $33 + x = 35$ and $x = 2$ .
Two games only: F and V only $9 - 2 = 7$ , F and H only $6 - 2 = 4$ , V and H only $11 - 2 = 9$ .
One game only: football $18 - 7 - 4 - 2 = 5$ , volleyball $20 - 7 - 9 - 2 = 2$ , handball $21 - 4 - 9 - 2 = 6$ .
Fill these into the diagram.

(b)(i)

All three: 2.

(ii)

Only one game: $5 + 2 + 6 = 13$ .

(iii)

Only football: 5.

(c)

Only two games: $7 + 4 + 9 = 20$ students, so $P = \frac{20}{50} = \frac25$ .

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