# My Favorite One-Liners: Part 62

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

This is a story that I’ll tell after doing a couple of back-to-back central limit theorem problems. Here’s the first:

The chances of winning a column bet in roulette is 12/38. The bet pays 2 to 1, meaning that if you lose, you lose $1. However, if you win, you get your$1 back and $2 more. If this bet is made 1000 times, what is the probability of winning at least$0?

With my class, we solve this problem using standard techniques with the normal approximation:

$\mu = E(X) = 2 \times \displaystyle \frac{12}{38} + (-1) \frac{26}{38} = - \displaystyle \frac{1}{19}$

$E(X^2) = 2^2 \times \displaystyle \frac{12}{38} + (-1)^2 \frac{26}{38} = \displaystyle \frac{37}{19}$

$\sigma = SD(X) = \sqrt{ \displaystyle \frac{37}{19} - \left( - \displaystyle \frac{1}{19} \right)^2} = \displaystyle \frac{\sqrt{702}}{19}$

$E(T_0) = n\mu = 1000 \left( -\displaystyle \frac{1}{19} \right) \approx -52.63$

$\hbox{SD}(T_0) = \sigma \sqrt{n} = \displaystyle \frac{\sqrt{702}}{19} \sqrt{1000} \approx 44.10$

$P(T_0 > 0) \approx P\left(Z > \displaystyle \frac{0-(-52.63)}{44.10} \right) \approx P(Z > 1.193) \approx 0.1163$.

Next, I’ll repeat the problem, except playing the game 10,000 times.

The chances of winning a column bet in roulette is 12/38. The bet pays 2 to 1, meaning that if you lose, you lose $1. However, if you win, you get your$1 back and $2 more. If this bet is made 10,000 times, what is the probability of winning at least$0?

The last three lines of the above calculation have to be changed:

$E(T_0) = n\mu = 10,000 \left( -\displaystyle \frac{1}{19} \right) \approx -526.32$

$\hbox{SD}(T_0) = \sigma \sqrt{n} = \displaystyle \frac{\sqrt{702}}{19} \sqrt{10,000} \approx 139.45$

$P(T_0 > 0) \approx P\left(Z > \displaystyle \frac{0-(-526.32)}{139.45} \right) \approx P(Z > 3.774) \approx 0.00008$.

In other words, the chance of winning drops dramatically. This is an example of the Law of Large Numbers: if you do something often enough, then what ought to happen eventually does happen.

As a corollary, if you’re going to bet at roulette, you should only bet a few times. And, I’ll tell my students, one Englishman took this to the (somewhat) logical extreme by going to Las Vegas and making the ultimate double-or-nothing bet, betting his entire life savings on one bet. After all, his odds of coming out ahead by making one bet were a whole lot higher than by making a sequence of bets.