Adventures in Probability (or, Why I Did Not Gamble in Reno)

As you might surmise, I made it back safe and sound from Reno last night. I had a good time at the conference, but I am very glad to be home.

Reno (like the rest of Nevada, I imagine) is chock full of slot machines, video poker, keno games, and other opportunities to gamble. There were slot machines in the airport terminal. There are video gambling consoles at the bar in many restaurants. They had keno runners in the restaurants, so you didn't have to interrupt your eating in order to gamble. The cost of lodging is relatively cheap, because it is subsidized by the gambling in the casinos.

The casinos felt incredibly depressing to me. They are designed for optimal gambling output, I imagine. They have no windows and the lighting makes it feel like it's the middle of the night, even at noon. You lose all sense of the passing of time.

I am of the opinion that the lottery is a tax on stupidity and desperation. Sure, somebody wins, but the probability of that person being you is lower than the probability that you will get struck by lightning. Similarly, I don't see the appeal of gambling games of chance. Maybe it is because I am not much of a risk-taker. Sure, I take risks, but I mentally calculate the probability of success before I take the jump. And all I have to do is take one look at the opulence of the companies that run casinos to know that gambling is a risk not worth taking!

As I was eating breakfast yesterday, I was offered the "opportunity" to play $1 Keno. In this "game," you pick any three numbers from 1 to 80. The casino draws twenty numbers from that same range, and if your three numbers were all drawn, you can win $35 and a $15 meal voucher.

I did a quick back-of-the-envelope calculation of my odds. Roughly speaking, there is a 1 in 4 chance that any number I pick will be drawn. So the chances of all my numbers being drawn is roughly (1/4)³ = 1/64 ≈ 1.56%. I knew that this calculation was not exact, but it gave me enough information to know that this game was set up in favor of the house.

Every time you play, you pay them a dollar. Once in 64 games, they have to give you $35 plus the meal voucher. If you spent the entire thing, that's still only $50 that they've given out, whereas they've taken in $64, for a net profit of $14.

The exact odds are actually lower. Here's how to compute them:

Let's suppose that you're picking numbers in the range 1-80 and that the casino has already drawn their numbers. The chance that the first number you pick is one that they drew is 20/80, because there are 20 numbers that they drew out of a total of 80. Let's suppose that you got that one right. What are your chances of getting the next one right? Well, there are now 19 possible right numbers out of a total of 79 numbers (because we've picked one of the right numbers and we can't pick it again). So the probability that we got this one right is 19/79. Similarly, if we have been lucky so far and are picking the third number, the probability that we pick another drawn number is 18/78 (because we've picked two of the right numbers and we can't pick them again). So the exact odds are 20/80*19/79*18/78 = 6840/492,960 ≈ 1.39%.

I wrote a little program to illustrate how this keno game works. It's short enough that I'll include my source right here.

#include <stdio.h>
#include <math.h>

int main() {
  long i = 0, ngames = 10000000, nwins = 0, npicked2 = 0, npicked1 = 0;
  int nmatched = 0, ndrawn = 20, nmax = 80, rand0, rand1, rand2;

  for (i = 0; i < ngames; i++) {
    rand0 = rand()%nmax + 1;
    do {
      rand1 = rand()%nmax + 1;
    } while (rand1 == rand0);
    do {
      rand2 = rand()%nmax + 1;
    } while ((rand2 == rand0) || (rand2 == rand1));
    nmatched = ((rand0 <= ndrawn) + (rand1 <= ndrawn) + (rand2 <= ndrawn));
    if (nmatched == 3) {
      nwins++;
    } else if (nmatched == 2) {
      npicked2++;
    } else if (nmatched == 1) {
      npicked1++;
    }
  }
  printf("Out of %ld games, you won %ld of them, or %f percent\n", ngames, nwins, (100.0*nwins)/ngames);
  printf("You picked exactly two numbers correctly %ld times, or %f percent\n", npicked2, (100.0*npicked2)/ngames);
  printf("You picked exactly one number correctly %ld times, or %f percent\n", npicked1, (100.0*npicked1)/ngames);
  printf("You picked no numbers correctly %ld times, or %f percent\n", (ngames-nwins-npicked2-npicked1), (100.0*(ngames-nwins-npicked2-npicked1))/ngames);
  printf("You paid %ld dollars, and won back %ld dollars\n", ngames, 50*nwins);
  printf("For a profit of %ld dollars (for the casino)\n", ngames - 50*nwins);
  return 0;
}

And here's the output of my program:

theano:~/misc rebecca$ gcc -o keno keno.c -lm
theano:~/misc rebecca$ ./keno
Out of 10000000 games, you won 139116 of them, or 1.391160 percent
You picked exactly two numbers correctly 1389265 times, or 13.892650 percent
You picked exactly one number correctly 4307469 times, or 43.074690 percent
You picked no numbers correctly 4164150 times, or 41.641500 percent
You paid 10000000 dollars, and won back 6955800 dollars
For a profit of 3044200 dollars (for the casino)

So as you can see, if I play keno ten million times, I pay ten million dollars and earn back nearly $6.96 million (including the meal voucher), resulting in a profit of over $3 million for the casino. So they earn a 30% profit on my "purchase" of keno.

This was the simplest game that they offered, but it is illustrative of all the games in a casino. You can see why I did not play any games of chance.

For keno, they draw actual balls. I wonder what sort of random number generation techniques they use for the video gambling games. I am curious to know, because it might be possible to take advantage of flaws in the random number generation. I've also read about people who used a computer program to defeat roulette. They used their cell phone to gather information about the current velocity and deceleration of the wheel and compute in which quadrant the wheel will stop. They had enough success at it that they made over £1.2 million. Not bad for a night's work!

Without fancy equipment like those three had at their disposal, your chances of overcoming probability are slim at best. My advice: don't bother. Keep your money and spend it on something else.