<s>[INST] Picking socks probability proof I'm trying to follow the proof from question 1b out of the book "50 Challenging Problems in Probability" by Mosteller. The problem states:
A drawer contains red and black socks. When two socks are drawn at random, the probability that both are red is $1/2$. How small can the number of socks in the drawer be if the number of black socks is even?
The answer given starts as follows:
Let there be $r$ red and $b$ black socks. The probability of the first socks' being red is $\frac{r}{r+b}$ and if the first sock is red, the probability of the second's being red now that a red has been removed is $\frac{r-1}{r+b-1}$. Then we require the probability that both are red to be $\frac{1}{2}$, or $$\frac{r}{r+b} \times \frac{r-1}{r+b-1} = \frac{1}{2}$$
Notice that 
$$\frac{r}{r+b} > \frac{r-1}{r+b-1} \quad \text{for b} >0$$
Therefore we can create the inequalities 
$$\left(\frac{r}{r+b}\right)^2 > \frac{1}{2} > \left(\frac{r-1}{r+b-1}\right)^2$$
This is where I'm confused. Why is it that $\left(\frac{r}{r+b}\right)^2 > \frac{1}{2}$? If $r=1, b=100$ then obviously $\left(\frac{1}{101}\right)^2 < \frac{1}{2}$. Am I missing some obvious assumption?
 [/INST] An explicit way to see the inequality is as follows. Start with the inequality between the multiplied fractions
$$\frac{r}{r+b}>\frac{r-1}{r+b-1}$$
Multiply both sides by $\frac{r}{r+b}$:
$$\left( \frac{r}{r+b}\right)^2>\frac{r}{r+b}\times\frac{r-1}{r+b-1}$$
The right hand side of the inequality is just $\frac{1}{2}$ from the initial probability equation. So
$$\left( \frac{r}{r+b}\right)^2>\frac{1}{2}$$
Similarly, if you multiplied the original inequality by $\frac{r-1}{r+b-1}$, you would get:
$$\frac{r}{r+b}\times\frac{r-1}{r+b-1}=\frac{1}{2}>\left(\frac{r-1}{r+b-1}\right)^2$$
Combine the two inequalities so obtained to conclude
$$\left( \frac{r}{r+b}\right)^2>\frac{1}{2}>\left(\frac{r-1}{r+b-1}\right)^2$$
</s>