We consider all distinct rational numbers
‌ such that
p,q∈{1,2,3,4,5,6} and GCD
(p,q)=1Now, we will consider each possible
q from 1 to 6 and for each, count the
p∈{1,2,3,4,5,6} such that g.c.d.
(p,q)=1 So, there are 22 distinct reduced rational numbers of the form
‌.
As we know that, a proper fraction is one where p
Among the 22 reduced fractions, we will now count how many satisfy p
When,
q=1;p=2,3,4,5,6⟶ all
p>q⟶ None proper fraction.
when
q=2,p=1,3,5⟶ only
p=1⟶1 proper fraction.
When
q=3;p=1,2,4,5⟶p=1,2⟶2 proper fraction.
when
q=4;p=1,3,5⟶p=1,3⟶2 proper fraction when
q=5;p=1,2,3,4,6⟶p=1,2,3,4⟶4 proper fraction when
q=6,p=1,5⟶p=1,5⟶ both
<6⟶2 proper fraction
∴ Total proper fraction
=1+2+2+4+2=11So, required probability
‌=‌| ‌ Number of proper fraction ‌ |
| ‌ Total distinct reduced fractions ‌ |
‌=‌=‌