Alex's Adventures in Numberland - Alex Bellos [132]
The law of large numbers says that if a coin is flipped three times, it might not come up heads at all, but flip it three billion times and you can be pretty sure that it wil come up heads almost exactly 50 percent of the time. During the Second World War, the mathematician John Kerrich was visiting Denmark when he was arrested and interned by the Germans. With time on his hands, he decided to test the law of large numbers and flipped a coin 10,000 times in his prison cell. The result: 5067 heads, or 50.67 percent of the total. Around 1900, the statistician Karl Pearson did the same thing 24,000 times. With significantly more trials, you would expect the percentage to be closer to 50 percent – and it was. He threw 12,012 heads, or 50.05 percent.
The results mentioned above seem to confirm what we take for granted – that in a coin flip the outcome of heads is equally likely as the outcome of tails. Recently, however, a team at Stanford University led by the statistician Persi Diaconis investigated whether heads really are as likely to show up as tails. The team built a coin-tossing machine and took slow-motion photography of coins as they spun through the air. After pages of analysis, including estimates that a nickel will land on its edge in about 1 in 6000 throws, Diaconis’s results appeared to show the fascinating and surprising result that a coin will, in fact, land on the same face from which it was launched about 51 percent of the time. So, if a coin is launched heads up, it will land on heads slightly more often than it will land on tails. Diaconis concluded, though, that what his research really proved was how difficult it is to study random phenomena and that ‘for tossed coins, the classical assumptions of independence with probability are pretty solid’.
Casinos are all about large numbers. As Baerlocher explained, ‘Instead of just having one machine [casinos] want to have thousands because they know if they get the volume, even though one machine maybe what we call “upside-down” or losing, the group as a collective has a very strong probability of being positive for them.’ IGT’s slots are designed so that the payback percentage is met, within an error of 0.5 percent, after ten million games. At the Peppermill, where I was staying during my visit to Reno, each machine racks up about 2000 games a day. With almost 2000 machines, this makes a daily casino rate of four million games a day. After two and a half days, the Peppermill can be almost certain that it will be hitting its payback percentage within half a percent. If the average bet is a dollar, and the percentage is set to 95 percent, this works out to be $500,000 in profit, give or take $50,000, every 60 hours. It is little wonder, then, that slots are increasingly favoured by casinos.
The rules of roulette and craps haven’t changed since the games were invented centuries ago. By contrast, part of the fun of Baerlocher’s job is that he gets to devise new sets of probabilities for each new slot machine that IGT introduces to the market. First, he decides what symbols to use on the reel. Traditionally, they are cherries and bars, but they are now as likely to be cartoon characters, Renaissance painters, or animals. Then he works out how often these symbols are on the reel, what combinations result in payouts, and how much the machine pays out per winning combination.
Baerlocher drew me up a simple game, Game A opposite, which has three reels and 82 positions per reel made up of cherries, bars, red sevens, a jackpot and blanks. If you read the table, you can see that there is a or 10.976 percent chance of cherry coming up on the first reel, and when this happens $1 wins a $4 payout. The probability of a winning combination multiplied by the payout is called the expected contribution. The expected contribution of cherry-aning-anything