One of the local newspapers has a puzzle with the prize of a 2 GB iPod Nano. You send an SMS with the solution to a number, pay 5 DKK (app. $1, a little less than 1€) and take part in the draw if your solution is correct.
I recently noticed that they are kind enough to print the number of correct solutions sent to them the previous week. It turns out it is very few. This week it was 28. That probably says something about the quality of the newspaper...
Anyway, what's interesting here is that you pay 5 kroner to get about 1/28 chance of winning a prize worth 1600 kroner. Assuming a probability of 1/28 every week, the outcome is binomially distributed. The expected number of wins is thus np, the number of draws times the probability. Consequently, after 28 weeks you can expect to have won the prize once, paying 140 kroner to get an iPod!
So we've decided to start sending in the solutions from now on.
The thing about gambling is that the variance is usually high. For the binomial distribution the variance is np(1-p). Since the probability is 1/28, the extra factor compared to the expected value is 1-1/28, or almost 1. In other words, the variance is almost the size of the expected value. We take the square root to get the standard deviation, but this does not change the value much.
For the normal distribution, which the binomial distribution approximates, one should not be surprised to get a value within 2 standard deviations of the mean (this is about 95% of the probability mass). Here are two plot that shows the expected value (red line) with one and two standard deviations added as a function of the number of weeks. The first one is zoomed in on 0-28 weeks.
The good news is that after 28 weeks, I wouldn't be surprised to have won two or even three iPods. The bad news is that I might as well not have won one at all. As you can see on the second plot, it takes more than 150 weeks to make the expected value minus two standard deviations greater than one.
I am crossing my fingers.