Marge was no mathematician, but she knew she had just discovered a foolproof system to get rich paying roulette.
She had been observing the spin of the wheel at the casino for several days. During this time she had noticed that it was surprisingly normal for there to be a sequence of spins when the ball fell into only black or only red slots. But five in a row of the same color was very unusual and six in a row happened only a couple of times a day.
This was going to be her system. The chances of the ball falling into a slot of the same color six times in a row were tiny. So, she would watch, and once it fell into, say, red, five times in a row, she would bet that the next one would be black. She was bound to win more often that she lost because six in a row was so rare. She was so confident that she had already started to think about how she would spend the money.
Baggini., J., The Pig That Wants to Be Eaten, 2005, p. 16.
As admitted, Marge is no mathematician. And she doesn't have a very good understanding of causality either. Previous spins of the wheel have no effect on future spins, so the odds of winning any one bet revert exactly to their mathematical calculation each time the table boss picks up the roulette ball. On an American table, with 2 green slots for the house and 16 each for red and black, that means a 5.3% chance the house will win and a 47.4% chance a bet on one of the colors will win. Just to see this in action, I spent a bit of time this morning putting together a spreadsheet that picked random numbers and awarded wins exactly like a roulette table. Then I ran the calculations up to 10,000 times. Here's what I saw:
No. Times Same Win In A Row
No. House Even Odd 2 3 4 5 6
50 3 26 21 20 8 3 1 0
100 8 48 44 42 18 5 3 1
250 16 121 113 102 47 22 12 7
500 28 239 233 246 125 61 30 14
1,000 65 445 490 466 225 109 60 33
2,500 151 1175 1174 1144 544 263 126 62
5,000 255 2401 2344 2301 1105 522 249 132
10,000 543 4717 4740 4440 2047 938 431 202
When I calculated the percentages of wins here, I saw that they converge almost perfectly to the mathematical odds predicted. Note too how every streak gets cut almost perfectly in half as it continues growing. I didn't count how many streaks were busted by house wins, but if Marge always bet the 5's in a row would end on the opposite color, she would win slightly less than 50% of the time most of the time she played this way, exactly like everyone else. It was fun to see that the longest streak in 10,000 spins was actually 13 in a row too, so if Marge had happened to try her scheme during that streak, she'd have lost 8 times in a row!
Exercises like this are good to go through every once in a while since our brains can understand the math, but our emotions sometimes lead us to believe otherwise. We haven't evolved very good "gut" reactions to questions of probability as has been shown in many different experiments. Whenever an outcome has a highly emotional positive or negative payoff, our rational responses are easily skewed.
She might randomly win a bit more or lose a bit more than average, but that also is just like everyone else at the casinos.