I have a probability problem book and one problem looks at the solution of the following:
a drawer contains red socks and black socks. when two socks are drawn at random the probability that both are red is 0.5. how small can the number of socks in the drawer be? how small if the number of black socks is even?
the solution solves for the general case with the following with r red socks and b black socks:
1. r/(r+b) x r-1/(r+b-1)=0.5
then sets up the inequalities
2. (r/(r+b))^2 > 0.5 > (r-1/(r+b-1))^2
taking the square root and multiplying by (r+b) gives first inequality as
3. r > 1/(2^0.5) x (r+b)
and the next step takes it to:
4. r > 1/(2^0.5 - 1) x b
I do not follow why we are able to go from 3 to 4. can anyone help out please?