Image by Vivek Kumar The Question A couple chooses to keep having children till they have an equal number of boys and girls. How many children will they have on average? In this article we will cover 2 ways to solve this problem. First using a random walk and proof by contradiction. The second is Optiver’s solution to this problem. Solution 1 Let $X_t$ denote the random variable for which child has been born at time $t$.

Image by sq lim Today’s article will look at one of the more difficult areas in probability called martingale theory. We will cover the basic theory and look at 2 examples which use martingale theory (the drunk man and the ABRACADABRA problem). The Theory Before jumping into the problem, we have to cover some of the theory behind martingales. It is important to mention we will be skipping over many of the measure theory definitions and theorems. These will be covered at a later time. If you wish to read up more about this area of mathematics, I’d recommend reading the paper by T. Smith here. This paper was the basis for this article [1].