The Petersburg paradox
In casinos worldwide people can play roulette with a so-called 'evens' or 'even money' bet.
Upon winning, such a bet pays out double the amount wagered, so the potential gain equals the amount wagered.
A player makes such a bet by putting money on either red or black. Since a roulette wheel
has both 18 red and black numbers, the likelihood of winning by betting or either red or black
are even: red: 18/36; black: 18/36.
If we assume a 'true;' i.e., unbiased roulette, and we record the number of times that red and black
turn up over a series of turns, we end up with two numbers for each:
- The so-called absolute frequency: this is the actual count of red and black turning up.
- The so-called relative frequency: this is the percentage of times of red and black turning up.
These two frequencies seem sometimes to be at odds with each other and confuse some people, especially people
who keep losing their money at the roulette table. Here is their (faulty) reasoning:
Darn, I have now bet on red a number of times and I keep losing. Still, since red and black each must show up
50% of the time in the long run —relative frequencies for a true roulette are 50% for each— red must eventually turn up,
and all I have to do is keep playing and increase my bet in order to win.
Yet, those same people know and realize that a roulette table has no memory and that it will therefore not be able
to compensate in the future for 'mistakes' it made in the past.
This apparent paradoxical situation is known as the
Petersburg paradox. and it is based on a misunderstanding of the concept of relative frequency.
Consider the following table:
Games
played | Red
count | Black
count | Count
difference | %
difference |
| 100 | 45 | 55 | 10 | 10 |
| 1000 | 490 | 510 | 20 | 2 |
| 10000 | 4,900 | 5,100 | 100 | 1 |
| 100000 | 49,900 | 51,100 | 200 | .2 |
| 1000000 | 499,500 | 500,500 | 1000 | .1 |
| ∞ | ∞ | ∞ | ∞ | 0 |
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In other words: whereas the relative frequency (%) gets ever closer to 50%, the difference
in the actual counts becomes ever larger!!
The bottom row in the table pushes this pattern to the extreme: if we would play an infinite number of roulettes,
the relative frequency of both red and black would become 50%, even though the difference between
the number of times that red and black would turn up would become infinitely large!
In yet other words: do not fall for the Petersburg paradox.
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Below you can play with this.
Two things to note:
- The preset value of n is set to 7. This means that 107
(= 100,000,000) roulettes will be simulated. For every 1 unit of n more, your browser will have
to compute 10 times as many roulettes. This puts some real strain on your browser. We ran a little
test with n = 9 on both the Chrome and Firefox browsers. Chrome took a little over a minute to
complete; Firefox did it in 20 seconds. Still, if you crank up n above 7, expect to have to wait
for the result.
- Because of the scale differences in plotted values, the plot's axes are logged.
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