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
q
Valid p (C.G.D. (p,q)=1
Count
1
2,3,4,5,6
5
2
1,3,5
3
3
1,2,4,5
4
4
1,3,5
3
5
1,2,3,4,6
5
6
1,5
2
Total
22
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 =