There's been a lot of discussion lately about tolerances on chip thickness, so I did a little analysis:
Let's call a perfect chip 0.131" thick.
Claybuster measured his ASMs as having a thickness range from 0.124" to 0.138", so I'll use those numbers.
Chip Thickness:
I don't have enough data to know the exact distribution of chip thicknesses. Claybuster's numbers might imply a gaussian distribution, but I'll assume a worse situation of a uniform distribution. So if I pick 10,000 chips at random, I might get a thickness distribution like this:
(Figure1 assumes all thickness between 0.124" and 0.138" are equally likely)
Stack Height:
So now I pick 20 of those chips at random and stack them up. Let's do that 1 million times. How tall will the various stacks be?
Figure 2 shows the (expected) gaussian distribution of the expected height of a stack of 20 chips.
Note that on average (and the most likely result, as well), the stack is 20 chips tall. The theoretically smallest stack would be 18.93 chips, but in 1 million trials, the shortest stack observed was 19.34 chips.
Similarly, the theoretically largest stack would be 21.07 chips, but in 1 million trials the tallest stack observed was 20.67 chips.
Note that the chances of pulling 20 chips that are each at the extreme of thickness (either thickest or thinnest) is vanishingly small. (That's why KENO is basically a sucker bet, because they don't offer the enormous payouts you'd need to hit these vanishingly small probabilities.)
Comparing Stacks:
Ok, so now we've got 1 Million stacks of 20 chips, drawn at random from our set. Let's pick 2 of those stacks and compare their heights. Let's do this experiment 1 Million times. How do the heights of 2 stacks compare?
So the most likely result is that the two stacks will be the same height. But there are tails on this distribution that are larger than +/- .5 chip. Let's consider the comparison to fail if the two stacks are more than 1/2 chip difference in height.
In this case, 10,525 out of 1 Million stack height comparisons resulted in a height difference of more than 1/2 chip. That's a failure rate of 1.05%.
Suppose we compare shorter stacks?
Changing the stack height to 10 chips yields:
Comparing stacks of 10 chips.
In this case, the failure rate (chance that two stacks of 10 will differ by more than 1/2 chip thickness) is 0.022%, or 2 times out of 10,000
Comparing stacks of 5 chips yields:
Comparing stacks of 5 chips.
Now, the failure rate was 0%. In 1 Million trials, the stacks were always within 1/2 chip in thickness.
Bottom line: your chances of making a mistake when comparing stacks of 20 or 10 is pretty small, but if you can't afford to make a mistake, compare stacks of 5 and count those stacks. (Isn't that what casino dealers are trained to do?)
Note that this analysis had a chip tolerance thickness of +/- 0.007, with a uniform distribution. If you had +/- 0.005 with a gaussian distribution, comparing stacks would be even more reliable.
I don't really have time to make millions of stacks of actual poker chips.
Linear Mode
