Leibniz Wrong About Dice!

I was reading a book about Leibniz called “Selections” and then “I, Claudius” by Robert Graves. Emperor Claudius wrote a small pamphlet about throwing dice. Leibniz had a small section about dice odds—that he was using to illustrate some other matter.

 

Two dice references in two different books got me thinking.

 

Leibniz got something wrong! He wrote about someone throwing two six-sided dice and rolling an “11”, and he wrote that there was only one way to roll an “11”.

 

Technically, there are two ways to roll and “11”: you can either roll 5 & 6…but you can also roll 6 & 5. Each die has a different outcome.

 

TWO SIX-SIDED DICE WAYS TO THROW EACH NUMBER

2 = 1+1

3 = 1+2 or 2+1

4 = 1 +3 or 3+1 or 2+2

5 = 1+4 or 4+1 or 2+3 or 3+2

6 = 1+5 or 5+1 or 2+4 or 4+2 or 3+3

7 = 1+6 or 6+1 or 2+5 or 5+2 or 3+4 or 4+3

8 = 2+6 or 6+2 or 3+5 or 5+3 or 4+4

9 = 3+6 or 6+3 or 4+5 or 5+4

10 = 4+6 or 6+4 or 5+5

11 = 5+6 or 6+5

12 = 6+6

 

As you can see, there are more ways (easier) to roll a “7” than any other number. Lucky Seven! 

When Leibniz made his error, he stated there was only one way to roll an “11”. But as you see, there are two ways, which makes “11” easier to roll and “2” or “12”—and the same odds as rolling a “3”. That’s a pretty significant difference-that makes the chances or rolling an “11” 5.56% instead of only 2.78% which is double the odds.

Scribbled notes on an old library catalog card I was using as a bookmark:

Claudius was never as dumb as he pretended to be.

Probability is really just plain old chance

“Probability is really just plain old chance.”        -Michael Logusz

"Gefühl ist alles name ist schall und rauch."-Goethe's Faust

This ties in with my last post about 1 = 0.99... and "common sense", "normal thinking", etc. versus mathematical theory and terminology. 

Probability merely studies abstract models and not real life! Probability is better understood sometimes as “possibly” or “more than likely” or “usually”. Chance!

Games of chance (gambling). The probability that I would never win at the slot machine approaches zero. Am I (and every other tourist in Vegas) a guaranteed winner? Nope, but the way some people approach probability in mathematics would make it seem so.

If you flip a coin the chances of heads/tails if 50/50.

If you flip a coin 99 times and get heads 99 times, the chances that the next time you flip the coin it will also be heads is still 50/50. Math deals with ideas, not necessarily the real world. 

My "monster toned" 1922 penny that I recently bought (with free shipping!). It sure is a beaut! If I it the "chances" are that it'll land on heads is 50% theoretically speaking. I just flipped it three times and got three tails in a row :(

For instance mathematicians deal with lines a lot. Their lines have no depth, breadth, width, etc. Engineers deal with lines in the real world: if you make a 1” hole in something and try to put a 1” peg in it, it will NOT fit! Engineers need to make the peg slightly smaller than the 1” hole to get it inside.

A common problem with CAD/CAM design is that people design things the ‘math way’ and do not take into consideration the width of the line they drew with a pencil—or with the computer. The result: parts are too tight and can’t go together.

So if (as discussed in the last post) infinities of nothing can still add up to something is there something analogous with chances? Yes: quantum tunneling. In many of my posts I reference radioactive alpha emitters, usually AM-241. According to "regular" physics it's impossible for a slow/weak alpha particle to break past it's barrier and be emitted. Impossible. Hmmm...according to quantum physics it's insanely rare. An alpha particle would have to bounce around three times ten to the thirty-first power! Almost no chance but still a nontrivial possibility. Schroedin and Heisenberg came up with the answers, alternating fixed time or the basis on their related equations...but that doesn't matter because of you read my other posts you'll see that I have a tiny piece of Americium-241 that blasts out so many "ultra-rare" particles every second of enough to irradiate old silver dimes and cause an amazing display in a radioscope (see later post "So you wanna see atomic particles with your own eyes huh? Part II").

Yes, something so rare it's impossible in regular physics but can easily be seen with some zinc-sulfide smeared on an eyepiece (radioscope / spinthariscope). Don't believe your eyes? Irradiate an old silver dime with AM-241 wrapped in aluminum foil: your Geiger counter will let you know you've done " The Improbable " a few zillion times over.