20.1 Discrete random variables

Let’s play a simple game. I toss a coin, and if it comes up heads, you win $1 . If it is tails, you lose $2 .

Up until now, we were dealing with questions like the probability of winning. Say, for the coin toss, whether you win or lose, we have

Despite the equal chances of winning and losing, should you play this game? Let’s find out.

After n rounds, your earnings can be calculated by the number of heads times $1 minus the number of tails times $2 . If we divide total earnings by n, we obtain your average winnings per round. That is,

where #heads and #tails denote the number of heads and tails respectively.

Recall the frequentist interpretation of probability from Section 18.2.7? According to our intuition, we should have

This means that if you play long enough, your average winnings per round is

So, as you are losing half a dollar per round on average, you definitely shouldn’t play this game.

Let’s formalize...