OK class, time for MATH!

Two odds multiplied is still odd. The odds of choosing an odd number at random is exactly 50%of course. What percent (so what are the odds of choosing one at random) of decimal expansions of fractions end in an odd digit, like 1/2=.5, 1/8=.125 but 1/5=.2 an even digit?

50%
It's an even chance.

I have not had time to think about it, can you prove it?

Unless one side is more weight,(which I heavily doubt, since they are stamped), the odds are even.
The machines that stamp them and the automated line they go through would necessitate equal weight distribution on both sides of the substrate. - (I believe it's still copper in quarters)

Ha,ha! Who can prove chance Ken.
Good one, you got me.

Now I had some time. Let's count a few. Here is a list in order by denominator of the fractions whose decimals terminate in either an even or odd digit.

Odd: 1/2, 1/4, 3/4
Even: 1/5, 2/5, 3/5, 4/5 even is now ahead by 1. Probability of an odd one 3/8.
Odd: 1/8, 3/8, 5/8, 7/8, 1/10, 3/10, 7/10, 9/10, 1/16, 3/16, 5/16, 7/16, 9/16, 11/16, 13/16, 15/16, 1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20 odd is now ahead by 23. Prob. odd 27/41.
Even: 1/25, 2/25, 3/25, 4/25, 6/25, 7/25, 9/25, 11/25, 13/25, 17/25, 19/25, 21/25, 23/25 odd is still ahead by 10. Prob. odd 27/53.
Odd: the following numerators over 32: 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31; the following over 40: 1,3,7,9,11,13,17,19,21,23,27,29,31,33,37,39. Odd is now ahead by 42. Prob. odd 59/85.
Even: over 50: 1,3,7,9,11,13,17,19,21,23,27,29,31,33,37,39,41,43,47,49. Odd ahead by 22. Prob. odd 79/105.
Odd: over 64: all the odd numbers, 31 in all. Odd ahead by 55. Prob. odd 110/136.
Even: over 80:32 numbers. Odd ahead by 23.
Odd: over 100, 40 numbers. Odd ahead by 63. Prob. odd 196/222.

Note the trend, the percentage of odd numbers in the stack is 3/8 or about .4, then 27/41 or about .6, 27/53 or .5, 59/85 or .7, 110/136 or .8, 196/222 or .9.Odds seem to keep climbing as we count higher, even though they do slip back sometimes.

The next two additions are 125 which adds 104 evens to the stack, and 128 which adds 64 odds. 160 then adds more odds. Odds seem to dominate. Do you still think it's 50-50?

What was the question? Two out of three?
All bets are off then.
Two heads, one tails? - I'll trust your calculations.
- - I jumped to an erroneous conclusion.
- - - From a faulty premise.
- - - - Why is this important?

It's not important. It will not affect your position in the Kingdom at all. But it is a pleasant diversion, as you have pointed out.

You know Ken, it is interesting how the odds are way over two out of three.
How did you find that out?
Calculous?
I stunk at higher math, I saw pictures the whole time they were explaining it.

No, Rick, I just started playing with the numbers. The only fractions that terminate have denominators divisible by only 2 and 5 (anything else keeps leaving a remainder when you divide into 10, which is what you get when you add a decimal point and zeros to turn the fraction to a decimal). If it is divisible by 2 and 5 in a pair, that makes 10, and dividing by 10 just eliminates a zero when you divide out to find the decimal. If it has more 2's not to make a pair, it comes out odd, because 2 into 10 ends in 5, which is odd. If it has more 5's, it comes out even because 5 into 10 is 2 which is even. Since 2 is smaller than 5 you will get more cases with more 2's than cases with 5's. That meant most are odd.

When I tried counting them, I ran into some problems, caused by the interactions of 2's and 5's, so I started counting them to get an estimate, and hence the earlier post. I'm not completely convinced yet, but, since 1-10 contains one 5, and 4 even numbers, I think it's odds 4/(4+1) or 80%. We don't count the 10, since it is new digit place, and we deal with it thusly: Since the pattern repeats at every digit place, I think the total is 4/5+1/10 of 4/5+ 1/100 of 4/5, etc. That total at infinity is 8/9. The count seems to match that so far, but I'm not completely sure I'm right.

If you mean you saw pictures that the explanations suggested to you, that's what professional mathematicians do, so you do not stink at higher math.

Okay... I did 1/2 thru 1/9 in a free moment but Ken you've gone far beyond that! Impressive!

Alright Ken. .......We have 4 possibilities that I can see. (1/3 of 12)
Where do you get five and two?

I mean seven possibilities. (I am counting in binary)

I mean seven possibilities. (I am counting in binary)
I don't know - 3 to the 3rd is nine.
I'm lost on it.

You have to try it to see what I mean. Try 2/3 for instance: 3 divided into 2.000000 is.66666666666.... and it never ends, so we can't use that one for our count. The only ones that end are numbers whose prime factorization is all 2's and 5's, because no other number divides into 10, without a remainder. If the division is to end, you cannot have a remainder, so those are the numbers we count for the denominators.

So, you can use 4 as a denominator, since 4=2x2. You can use 20=2x2x5. You can use 100=2x2x5x5. You can't use 12=2x2x3, because of the 3. That's what prime factorization is, the product I wrote for each of them.

3 to the third is 27.

I cheated. I had a number theory class in college. We learned some of that there.

Ken are you a robot? or a calculator or something? where can I buy a pair of smarty pants?

I'm a mathematics professor, Blain. You know that. This is my career and my job.

Ya I know I just like making Jokes:]

3 cubed is 27, of course!
Ken, why can't we just round numbers. - Why not 2/3 = .667?
Would that make a huge difference?
So 2 and 5 are denominational factors that reduce to a quotient.

I see.
2 and 5 are the only numbers that factor into 10.
It takes me awhile...but once I get it, I latch onto it like a pitbull.
You have an interesting vocation Ken.