This is the solution of Problem #1.4 from Probability Interview Questions.

Problem statement

There are two balls in a bag. I tell you that at least one of them is red. What is the probability that both are red?

Answer

This is a simple counting problem. We're given 2 balls, each can be either red or not red.

With two balls like this, the set of all possible outcomes contains 4 outcomes that look like this:

Sample space - all possible outcomes with two balls which are either Red or Not Red
Sample space - all possible outcomes with two balls which are either Red or Not Red

Since we're told that at least one of the balls is red, our problem deals with a conditional universe:

Conditional sample space (or conditional universe), where at least one ball is Red
Conditional sample space (or conditional universe), where at least one ball is Red 

In this conditional universe, where at least one ball is red, there are 3 possible outcomes instead of 4:

$$\left\{\begin{align*}\left(Red, Red\right)\\\left(Red, \neg Red\right)\\\left(\neg Red, Red\right) \end{align*}\right\}$$

Since we're asking what's the probability of both balls being red given that at least one is red, there is only 1 good outcome, $(Red, Red)$, out of 3 possible outcomes.

Thus our probability is simply: $1/3$