@fiery : There are two parts to the proof .

Coins of type 3 don’t matter .

Initial draw doesn’t matter .
You can prove both the claims by mathematical induction
Proof 1 :
p(t1,t2,t3) = t1/(t1+t2+t3) + t3 / ( t1+t2+t3) * p(t1,t2,t31) . ( Statement 1 )
By induction hypothesis ,
p(t1,t2,t31) = t1/(t1+t2) .
which proves Statement 1 gives us assumed formula
Proof 2 :
p(t1,t2,t3,t4) = t1/(t1+t2+t3) * p(t11,t2,t3,t41) + t2/(t1+t2+t3) * p(t1,t21,t3,t41) + t3/(t1+t2+t3) * p(t1,t2,t31,t41) . (statement 2)
By induction hypothesis ,
p(t11,t2,t3,t41) = (t11)/(t11+t2)
p(t1,t21,t3,t41) = (t1)/(t1+t21)
p(t1,t2,t31,t41) = (t1)/(t1+t2)
which proves statement 2 gives us assumed formula .