Wikipedia:Reference desk/Archives/Mathematics/2009 February 14

= February 14 =

Mathematical fraction
The term "a third of a mil" in reference to a dollar amount is used in a New Jersey state statute. Please advise what that fraction is and what the decimal number is that should be used when multiplying another larger number. For instance what would I multiply $1,000,000 by to find a "third of a mil" of that amount?

Thank you,

Frank J. Mcmahon Mahwah, NJ [email removed] —Preceding unsigned comment added by 69.127.4.198 (talk) 02:07, 14 February 2009 (UTC)


 * Try a lawyer? I'd assume offhand "a third of a mil" means $$\frac 13 \cdot 1000000$$ dollars.  If there's some special legal meaning of the phrase, then it is something a lawyer would know, not a mathematician.  Maybe it would help if we could see the full sentence that "a third of a mil" first appears in.  By the way, we never email responses, so I removed yours to lessen visibility to spam-bots....  Eric. 131.215.158.184 (talk) 07:17, 14 February 2009 (UTC)


 * In some contexts I believe that, just like "1/3 per cent" means X / 300, this could mean "1/3 per thousand" or X / 3000. -- SGBailey (talk) 09:48, 14 February 2009 (UTC)


 * I would think "a third of a mil" is just short for "a third of a million" or $333,333.33. As SGBailey says, it could mean "a third per mil", or 1/3000 times. Either way, it's not a very common way of saying it, but then lawyers like to make things as confusing as possible - it keeps them in work. --Tango (talk) 12:53, 14 February 2009 (UTC)


 * I would assume that it refers to mill (currency). "A third of a mil" is 1/3000 of a dollar. -- BenRG (talk) 13:10, 14 February 2009 (UTC)


 * In light of, about library funding in New Jersey, it looks like SGBailey got it right; it's a third per mil. Of course, that also means it's a third of a mill per dollar (in this case, per dollar of assessed property value). So to answer the original question, for a property assessed at $1,000,000, a "third of a mil" would be around $333. — JAO • T • C 14:54, 14 February 2009 (UTC)

Numbers
I have these series of numbers and I know they are relate, but I don't know what they are called.

0,1,3,6,10,15, 21,28...

0,+1,+2,+3,+4,+5...

Thanks --68.231.197.20 (talk) 06:47, 14 February 2009 (UTC)


 * You should look at the triangular numbers. Eric.  131.215.158.184 (talk) 07:07, 14 February 2009 (UTC)
 * The OEIS is a good place to answer these questions. Algebraist 14:23, 14 February 2009 (UTC)

they're the perfect squares. 0*0=0, 1*1=1, 2x2=4, 3x3=9, etc. that is +1+3+5 etc. Do you notice a relationship with your series? How would you express that as an equation -- or maybe two? note: this may be BS on my part —Preceding unsigned comment added by 82.120.236.246 (talk) 21:59, 14 February 2009 (UTC)
 * What are the perfect squares? None of the numbers in the list (other than 0 and 1) are squares... what are you talking about? --Tango (talk) 00:02, 15 February 2009 (UTC)
 * Perhaps the sum of any two consecutive terms;) hydnjo talk 13:56, 15 February 2009 (UTC)

Dice game problem
I was thinking about a problem this morning and got a bit stuck on figuring out how to calculate the answer. It seemed like the sort of problem that has been calculated before, but I can't seem to find anything on it. Here's how it works:

You are playing a dice game with three six-sided dice. You roll the dice and set aside any that come up 1. You then reroll any dice that aren't one and continue the process of rerolling and setting aside 1's until all three dice have come up 1. The question is how many times on average do you need to roll the dice before they all individually have come up 1?

(Note: The original problem I'm actually trying to solve is very similar, except that instead of the probability of individual success being 1/6 the probability of individual success is 1/11.)

My friend and I worked out a brute force way to approximate it by computer (I think, I haven't typed it up yet), but I'm wondering if there's a more elegant or exact solution. Any suggestions? Thanks for the help. 71.60.89.143 (talk) 20:54, 14 February 2009 (UTC)

Just a quick follow-up - I used Excel to calculate that the expected number of rolls would be approximately 4.878 in order to get at least one success on each of the three dice. For my original problem where the chance of success is 1/11, the expected number of rolls is 8.623. Please feel free to confirm if you like, and if you have a nifty way of solving the problem I'd be interested in reading it. 71.60.89.143 (talk) 22:56, 14 February 2009 (UTC)
 * Your answers are certainly wrong. The expected number of rolls of one dice until you get a 1 is 6 (or 11), and getting the right number on all three is clearly harder. Algebraist 23:03, 14 February 2009 (UTC)
 * For three dice, probability p of success each time, I get $$\frac{11-19p+12p^{2}-3p^{3}}{p(2-p)(3-3p+p^{2})}$$, giving 10566/1001 for probability 1/6 and 45727/2317 for probability 1/11. Unfortunately, all I have for the general case (n dice, probability p) is a messy infinite sum, and I don't have time right now to do this properly. Algebraist 23:10, 14 February 2009 (UTC)


 * Yeah, after I typed the above I realized I made a mistake in my formula in Excel. I'm still not getting the same answer you got above, though. I get 7.38 for the expected value of p=1/6. Hmmm....  71.60.89.143 (talk) 23:26, 14 February 2009 (UTC)


 * To clarify what I'm doing in Excel, I let P(k,n) be the probability that at least k of the three dice have succeeded (k=0 to 3) after at least n rolls. So P(2,5) would be the chance that at least two of the three dice hit a success after five rolls.  Given that definition for P(k,n), I get the following formula for P(3,n) for n>1.


 * P(3,n) = P(3,n-1) + (P(2,n-1) - P(3,n-1))*(P(1,1) - P(2,1)) + (P(1,n-1) - P(2,n-1))*(P(2,1) - P(3,1)) + (1 - P(1,n-1))*P(3,1)


 * Above, the expression P(2,n-1) - P(3,n-1) is the probability that exactly two of the dice succeeded after n-1 rolls. The other expressions are similar. 71.60.89.143 (talk) 23:46, 14 February 2009 (UTC)


 * Alright, I found the error in my Excel sheet and the formula above. :)  The second term in the last few products is a probability involving rolling all three dice, but it should actually be only partial rolls. I fixed the error and, lo and behold, it matches your answer Algebraist.  Good work!  71.60.89.143 (talk) 01:48, 15 February 2009 (UTC)

Just my take. Let X be the number of throws before you get 1 on an n-faced die. Its cdf is
 * $$P(X\leq x)=\frac 1n \sum_{i=0}^{x-1}\left(\frac{n-1}{n}\right)^i=1-\left(\frac{n-1}{n}\right)^x$$

If you try to throw k dice, it is equivalent to look at the maximum of k iid random variables, which has cdf
 * $$P(\max\{X_1,\ldots,X_k\}\leq x)=\left[1-\left(\frac{n-1}{n}\right)^x\right]^k:=F_{k,n}(x)$$

Thus the expected value of number of throws should be
 * $$E\,\max\{X_1,\ldots,X_k\}=\sum_{x=1}^\infty x (F_{k,n}(x)-F_{k,n}(x-1))=\sum_{x=1}^\infty\left(1-\left[1-\left(\frac{n-1}{n}\right)^x\right]^k\right)$$

which I am pretty sure is possible to calculate explicitly. (Igny (talk) 02:49, 15 February 2009 (UTC))
 * Yeah, that's the infinite sum I alluded to above. Algebraist 03:16, 15 February 2009 (UTC)
 * So if I did not screw up for 3 dice with 6 face the average number of throws is 10.56 and for 11 faces it is 19.74 (Igny (talk) 21:17, 15 February 2009 (UTC))
 * That's right, Igny. And thanks for breaking down your solution; it's a little simpler and more generalizable than what I used. 63.95.36.13 (talk) 14:56, 16 February 2009 (UTC)

is it possible to fuck up when combining two random number generators?
If you're combining two random number generators that you think are pretty random, but who knows, maybe every so often they aren't random enough, is there any way to do that which looks okay and basically has the correct distribution, but in fact now is way not so random? Thanks.


 * P.s. this isn't malicious! I wouldn't be asking it this way, and on this forum, if it were  —Preceding unsigned comment added by 82.120.236.246 (talk) 22:23, 14 February 2009 (UTC)


 * This isn't exactly an answer to your question, but: for the most part, any attempt to combine two good PRNGs, or to modify the output of a PRNG to make it "more random", will actually reduce the its quality. That's why there are so few of them that are considered cryptographically strong. By the way, if you do get a straight answer to your question, be sure to remember it in case you ever want to participate in the Underhanded C Contest. « Aaron Rotenberg « Talk « 23:16, 14 February 2009 (UTC)


 * Yes, it is possible to fuck up. By not defining what distribution you really need. Cuddlyable3 (talk) 23:39, 14 February 2009 (UTC)


 * If you can reduce the randomness of a PRNG by combining its output in some simple way with another PRNG, then it certainly wasn't cryptographically strong in the first place. However it is usually possible to improve the randomness of even weak PRNGs by combining them.  I flatly disagree with Aaron when he claims that it "usually" reduces the quality.  It can happen, if there is some unsuspected connection between the two PRNGs (or, of course, if you make some silly mistake in how you "combine" them), but it usually helps rather than hurts.
 * The simplest way to combine two PRNGs (say, normalized to return a value between 0 and 1) is simply to add their outputs modulo 1. If you do this to two PRNGs of relatively prime period, the period of the new PRNG is the product of the original periods.  (Period by itself is not a good measure of randomness, but short period is always a problem.)
 * An even better way is the McLaren–Marsaglia method, in which you cache values from one of the PRNGs, and use the other one to select a value from the stream. --Trovatore (talk) 23:52, 14 February 2009 (UTC)
 * "if you can reduce the randomness..by combining its output in some simple way ..then it certainly wasn't ..strong" WTF!! How's this for starters:
 * I suppose the result " " means that I just proved Perl's random number generator is way, way insecure!!! —Preceding unsigned comment added by 82.120.236.246 (talk) 00:58, 15 February 2009 (UTC)
 * I don't actually speak Pathologically Eclectic Rubbish Lister so I'm not quite sure what you're doing here. It looks like you're demanding that two random values chosen from 0 to 2 both be less than 1 in order to increment $zero, in which case I'd expect only 25% zeroes from a good RNG.  But I certainly wouldn't be surprised if Perl had a bad RNG — in fact, that seems more likely than not.
 * In any case you seem to have ignore both of my stipulations — that the two PRNGs be unrelated, and that you not make some silly mistake when combining them (like choosing random bits from a 25-75 proposition). --Trovatore (talk) 03:28, 15 February 2009 (UTC)
 * In any case you seem to have ignore both of my stipulations — that the two PRNGs be unrelated, and that you not make some silly mistake when combining them (like choosing random bits from a 25-75 proposition). --Trovatore (talk) 03:28, 15 February 2009 (UTC)


 * int rand 2 + int rand 2 in Perl means int(rand(2 + int(rand(2)))), which should be 0 with probability 5/12 ≈ 0.4167, so Perl's RNG passed this test—though I suspect it wasn't the intended one. -- BenRG (talk) 04:21, 15 February 2009 (UTC)
 * Yes, cryptologists have studied this problem in great detail and to use your vernacular, you can indeed fuck up badly this way. See for example Ueli Maurer and J. L. Massey, "Cascade ciphers: The importance of being first" and follow some of its references. 207.241.239.70 (talk) 06:41, 18 February 2009 (UTC)