Probability

We consider all distinct rational numbers ‌

p

q

such that p,q∈{1,2,3,4,5,6} and GCD (p,q)=1
Now, 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 ‌

p

q

.
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=11
So, required probability
‌=‌

‌ Number of proper fraction ‌

‌ Total distinct reduced fractions ‌

‌=‌

11

22

=‌

1

2