# IMAT 2012 Q77 [Probability]

In a group of students, exactly {\frac {2}{5}} are male and exactly {\frac {1}{3}} study mathematics. The probability that a male student chosen at random from the group studies mathematics is p.

Which of the following is the range of possible values of p?

A. {\frac{1}{3}\le p \le 1}
B. { 0\le p \le \frac {5}{6}}
C. {\frac{1}{3} \le p \le \frac{2}{5}}
D. {\frac{2}{5} \le p \le \frac{5}{6}}
E. {0 \le p \le \frac {1}{3}}

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TBC

The easiest claim we can make when we first read this question is the fact that none of the males could have been selected. This means the range of possible values for p must begin at 0.

these are not generally studied in school courses, and that is because this is a non-routine
question. It requires the use of problem solving strategies. In this question it is case of
considering ‘best’ and ‘worst’ case scenarios.
The solution to this problem involves taking the two extremes:

1. That the number of males studying maths is maximised
2. That the number of males studying maths is minimised
Considering case 1:
The maximum number of males studying maths. In this case we could have all the maths
students being male because 5
2
3
1
<
. Let’s say there are N students in total so that we are
dealing with numbers rather than probabilities, then in this case the number of male
students studying maths is 3
N
, and the number of male students is 5
2N
. The probability of
a student from the set of male students studying maths is then 6
5
2
5
35
2
3
=×=÷=
N
NNN
p
Considering case 2:
Now we want to minimise the number of males students studying maths. So this means
maximising the number of female students studying maths. 5
3
of the group of students are
female, so all the maths students could be female, because 5
3
3
1
<
. In this case none of the
male students study maths, p = 0 .
The actually probability can lie anywhere between and inclusive of these values:
6
5 0 p ≤≤ .