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September 30

Solving a modular diophantine equation

If I have an equation of the form ax+by=0(mod c), where a, b, and c are constant and pairwise relatively prime, how do I find all possible integral values of x and y? --75.50.49.19 (talk) 02:05, 30 September 2009 (UTC)[reply]

Since a is relatively prime to c, ax can take on every possible value mod c (why?). Likewise for by. So the problem is matching up each value of x with the correct value of y. First find y such that a1 + by = 0, and the rest of the solutions should follow from there. Rckrone (talk) 02:53, 30 September 2009 (UTC)[reply]

Details: since b and c are relatively prime by assumption, b has a multiplicative inverse β mod c (indeed, GCD(b,c)=1 is equivalent to bβ+cγ=1 for some integer numbres β and γ: and β is thus an inverse of b mod c). Now if x, y is any solution pair of your congruence, multiply by β and get y=-βax mod c. Conversely, the pair (x,-βax) clearly solves the congruence, for all x. --pma (talk) 17:38, 30 September 2009 (UTC)[reply]

Teminology request

I'm just looking for some terminology to aid in searching Google scholar articles... If you've seen Collecta, you know it is a real-time search engine. I'm interested in reading about the mathematical algorithms involved in real-time searching. Unlike Google/Bing/et al, these algorithms find objects to be popular for a short time and then go stale. I'm interested in seeing how to avoid having popular items dominate the search, keeping new items from getting in. For example, if I only have a database where I can hold 10 items and I have 10 popular items, I need a way to drop one for a new item that could be popular. Obviously, I don't know a thing about the proper terminology here, so my searches have been futile. -- kainaw ™ 13:13, 30 September 2009 (UTC)[reply]

Odds + probability

It's very nice that there are these two concepts, and I understand the basic difference, but who cares? What is the need to have both of them? DRosenbach ^{(Talk | Contribs)} 14:42, 30 September 2009 (UTC)[reply]

Some concepts are better represented as a probability. For example, what is the probability that a die will roll a 6? Some concepts are better represented as odds. For example, what are the odds that the Chiefs will break the Buccaneers' losing streak record? It also has a lot to do with the audience. If the audience is highly scientific, probabilities make more sense. If the audience is more general, odds make more sense. This is because a probability is basically the odds out of 100 tries. Odds limit the number of tries. So, I say you have 1 in 2 odds, you know that if you try two times, you are likely to see something happen. If I say you have 0.5 probability, you have to translate that to 1 in 2. -- kainaw ™ 14:49, 30 September 2009 (UTC)[reply]

If you say 1 in 2 odds, you really mean 1 in 2 probability. Odds are not stated that way. Odds would be stated as 2 to 1 against, which means there is 1 chance for it happening for every 2 chances for it not happening. So, actually odds is where you have to translate that 2 to 1 odds against actually means 1 in 3 probability. StatisticsMan (talk) 15:25, 30 September 2009 (UTC)[reply]

If you look at the article Odds, you can see one reason to have both. As I mentioned above, if I want to know how likely something is, saying it is 2 to 1 against does not make it immediately clear that the probability is 1/3. On the other hand, if I am betting, 2 to 1 odds against makes everything clear. If I bet $1, I will win $2 (and keep my original $1). Any way, I'm no expert and I'm not giving you THE correct answer. But, there are many parts of mathematics or any other subject where you have more than one way to describe something because one makes more sense sometimes and the other makes more sense sometimes. I find it harder to think in odds so I'd rather just stick to probability and I can do any problem with probability that I could do with odds, so that's fine. Maybe someone else is better with odds. StatisticsMan (talk) 15:50, 30 September 2009 (UTC)[reply]

Any way, I'm no expert and I'm not giving you THE correct answer. -- sort of makes me question your choice of usernames :) DRosenbach ^{(Talk | Contribs)} 03:32, 1 October 2009 (UTC)[reply]

I believe the name comes from being a statistics student, not a professional statistician. --Tango (talk) 14:22, 1 October 2009 (UTC)[reply]

Indeed - it is important to realise that odds and probability are more than just different notation. The confusion often comes from statements like "1 million to one" and "one in a million" - they aren't the same thing, but they are so close to being the same thing that it makes no difference (we're only working to 1sf anyway). For smaller numbers, the difference can be very significant. --Tango (talk) 16:26, 30 September 2009 (UTC)[reply]

(In response to Tango) Excuse my crassness, but how are they more than notationally different? Don't they represent the same data?--Leon (talk) 16:56, 30 September 2009 (UTC)[reply]

Yes, but in a very different way. "1 in 3", "1/3" and "0.333..." differ purely in notation, they are different ways of writing exactly the same thing. "2 to 1 against" means the same thing, but is clearly expressed in a different way - the numbers involved are different and a calculation is required to convert from one to the other. Perhaps my terminology isn't ideal, hopefully I've clarified what I mean. --Tango (talk) 17:09, 30 September 2009 (UTC)[reply]

Probabilities are easier to manipulate mathematically (we have simple rules like "probabilities always add up to 1"). Odds make it easier to calculate gambling winnings. I think that is the main reason for using each. --Tango (talk) 16:26, 30 September 2009 (UTC)[reply]

Although we need to be careful when writing 1/3. In a bookmaker's the notation m/n means m-to-n against. So 1/3 would mean 1-to-3 against, i.e. success is expected three times in four. Although that's not strictly true, a bookie always factors in his or her profit margins: for example, they might pay 10/11 (read as 10-to-11) for flipping a head, that way if you bet £1 on heads and £1 on tails you would only collect £1.91 and the bookie would have made 9p profit. ~~ Dr Dec (Talk) ~~ 21:39, 30 September 2009 (UTC)[reply]

Seems odd (haha!) to have a notation just so we can gamble. DRosenbach ^{(Talk | Contribs)} 03:28, 1 October 2009 (UTC)[reply]

Odds are usually integers, while probabilities are usually not integers. The possible outcomes of tossing three coins are 0, 1, 2, or 3 heads, and the corresponding odds are as 1 to 3 to 3 to 1. The probabilities are obtained by normalizing the odds, meaning dividing by the sum of odds 1+3+3+1=8, so the probabilities are 1/8, 3/8, 3/8, and 1/8. The odds are much easier to pronounce. Bo Jacoby (talk) 06:58, 1 October 2009 (UTC).[reply]

I'm afraid to say that odds are not at all usually integers. Take a look at the Ladbrokes website, for example this page. Chelsea are 6/5 to win the Premier League, Liverpool are 11/2. To be top goal scorer Fernando Torres is favourite at 6/4, Didier Drogba and Wayne Rooney are joint second favourite at 7/2. It's only the bigger odds, like 10/1 and above that are usually integers. Some common odds are 4/5, 10/11, 9/4, 9/2, 10/3, and oddly, 6/4 (they don't cancel the common factor of 2!) I used to work in a bookies when I was an undergraduate ;-) For our American editors: the LA Lakers are 11/5, the Cleveland Cavaliers are 7/2 and the Boston Celtics are 15/4 to win the NBA Championship 2009/2010. ~~ Dr Dec (Talk) ~~ 10:42, 1 October 2009 (UTC)[reply]

... and odds are are a more natural, intuitive notation than fractional probability, but the latter much more recent notation makes the arithmetic simpler. Dbfirs 08:07, 3 October 2009 (UTC)[reply]