An obvious example is in gambling. Gamblers often have this habit of finding illogical ways to improve their future (in terms of winning probabilities), which can often also be pure superstition, like avoiding being touched on the shoulders, or avoiding books, or somehow having lucky numbers somewhere in their possessions.
There is absolutely no link between having 8 lucky charms in your 8 pockets and the probability of winning in a game of roulette. Even so, it is fascinating to learn about the ways with which people try to improve their luck.
A more counterintuitive example would be trying to play on the slot machine that has not any winner for the longest time among the machines in a casino. Indeed, it is a seductive notion that such a machine would provide the greatest chance of a jackpot, or at least, a win.
This is wrong, because of 2 reasons.
First, such an improvement would be significant only when a large number of tries is involved. For player with a limited budget like mine, the number of tries will not exceed 100. This is not a large number. Even if it does exceed 100, it is very likely that the credits in that machine is very cheap, maybe 1 credit per cent. In this case, the payout will be very low.
Of course, if one was able to afford it, one could search for such a lot machine, and spend a whole day pressing the "Go!" button continuously. Ouch!
Second, even if previous players have been losing on a particular machine for 10000 consecutive rounds, the chance of hitting a jackpot for the 10001th round will still be the same.
At times, one may find that this second reason seems to be wrong. For example, you leave a machine after a long losing streak. Another player takes your seat in front of that machine and hits the jackpot. You think that you have just missed your chance, but that is not so.
Modern slot machines rely on a random number generator that produced a different combinations every second. What truly matters is the moment you play it. Therefore in this case, even if you had stayed for another round, the moment you press the button may not be the same as the one the winner used.
Poorly programmed machines have a tendency to repeat their combinations, but this a rare since it is easy to program a generator with periods that can last the whole lifetime of the Universe.
Like in coin-flipping, just because the last 4 flips have been heads doesn't make the 5th flip more likely to be tails.
Here's an extremely fascinating problem I found on wikipedia.
A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door." You begin by pointing to door number 1. The host shows you that door number 3 has a goat.
Do the player's chances of getting the car increase by switching to Door 2?
Everyone knows that in the circumstances described above, the chance of getting the car is 50/50, so changing doesn't matter.
Unfortunately, everyone's wrong here.
The fact is that a player doubles his chance of winning a car when he switches his choice, ie, by switching, his chance of opening the right door becomes 2/3.
Incredibly counterintuitive! The explanation is in the article, but I figured that if you're reading this you are probably are too lazy to click that link anyway.
The reason is simple. Here, we assume that the player will definitely switch his choice, and the host will definitely open a door that has a goat ( obviously he won't open the door with the car and still ask if the player wants to switch his choice of goat - "So will it be this old goat with the white hair or that old goat with the brown hair for you, sir?").
Scene 1: Man chooses A (Car), B (goat) opened, final choice is C (goat)
Scene 2: Man chooses A (Car), C (goat) opened, final choice is B (goat)
Scene 3: Man chooses B (goat), C (goat) opened, final choice is A (car)Scene 4: Man chooses C (goat), B (goat) opened, final choice is A (car)
Scene 5: Man chooses B (goat), C (goat) opened, final choice is A (car)
Scene 6: Man chooses C (goat), B (goat) opened, final choice is A (car)
3 is identical to 5, ditto for 4 and 6, because each door has an equal chance of being chosen at first by the player.
Amazing how things aren't always what they seem to be, isn't it?
In a casino, it is possible to win in the short term, but the odds in the long term are always in favour of the house. "House's advantage" isn't called that for nothing.
Another fascinating observation is how the dealers can do what they do ( dealing, counting, paying, etc) perfectly and never make a mistake. Do they have shifts with frequent breaks, or are these people really androids?
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