Talk:Lottery mathematics

Table of contents
I have added a section of calculating a score of n when selecting 6 from 49. Anyone know why the table of contents isn't showing, please?--New Thought 17:04, 3 March 2006 (UTC)


 * hmmmm... when it got to 4 sections, it seems to have added it automatically. — Preceding unsigned comment added by New Thought (talk • contribs)

Incorrect fraction
I believe the table under "Odds of getting other possibilities in choosing 6 from 49" is incorrect. The fractions listed are inverted. The $${49\choose 6}$$ should be in the denominator for each of those cases. The other numbers in the table are correct. — Preceding unsigned comment added by 128.84.216.236 (talk • contribs)


 * Whoever made the comment above was correct. I have made corrections to the table and the explanatory text. --Aaron Lehmann 11:47, 18 December 2006 (UTC)


 * Incorrect calculation: combin(42,0)=1 not 0. You need a section with calculation, code or calculator.
 * 42 nCr 0 = 1; combin(42,0) = 1;
 * This has a reason. Why is it wrong on the page?
 * There are three subsets for 6+/49 with lengths: 6; 1 and 42.
 * Chances for 5+1: 6 nCr 5 * 1 nCr 1 * 42 nCr 0.
 * There are more people ignoring big integers on wiki. They erase perfect pages, change good to wrong and so on. 2A02:A03F:E03F:6300:187A:5CC2:AB92:A706 (talk) 14:21, 6 February 2023 (UTC)

Wording
Good morning, I have only a basic college level course in statistics and probability, so it is not the equations in the article I am commenting on because the math is correct even if I think there is an easier way to explain it all, it is the explanation paragraph I think is not really correctly worded:

"Starting with a bag of 49 differently-numbered lottery balls, there is clearly a 1 in 49 chance of predicting the number of the 1st ball out of the bag. Accordingly, there are 49 different ways of choosing that first number. When the draw comes to the second number, there are now only 48 balls left in the bag (because the balls already drawn are not returned to the bag), so there is now a 1 in 48 chance of predicting this number."

My suggestion is that if the numbers do not have to be picked in order, there is not "clearly a 1 in 49 chance" of predicting the first number, but actually a 6 in 49 chance of picking the first ball correctly There is a 5 in 48 chance of picking the second correctly and so on. Only in the case of the balls being required to be picked in order would each selection have a 1 in something chance. 49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/1  wspruce@msn.com

The 49/6 x 48/5 x 47/4 x 46/3 x 45/2 x 44/1 Yields the number of combinations available (13,983,816). I am planning on adding a column in the table 1/Probability because that is what is on most play slips. —Preceding unsigned comment added by 198.190.9.225 (talk) 15:03, 9 June 2008 (UTC)

External link quality
the link to the online probability calculator should be removed or changed. the quality of the calculator is quite low, with most of the powerball calculators not even reporting correct results.

Myqlarson (talk) 07:11, 9 April 2011 (UTC)

0-99 no of lottry mathemetics
privious no (100-privi. no)+50 is that basic no then next no wich is coming suppose that 97 then privious no (-) ,(+) for ex.

privi.no is 114 then next no is 97

114         x   y  z  a   b   c          97 17 211 03 131 325 111 117 225                   u   k    r

b+.5(z-a)= ? b+.5c+.25(z-c)=?

Incorrect assumption
This aricle and the at least one of the references incorrectly assume that lotto numbers are not ordered. After being drawn, the numbers are put in order and therefore change the odds on each number slot! A statistical analysis of the lotto numbers drawn from any state lotto over the last 30 years shows a clear gausian for each 'slot'

ACorbs (talk) 14:07, 24 March 2010 (UTC)

Explanation needed
Hello. The portion "The numerator equates to the number of ways one can select the winning numbers multiplied by the number of ways one can select the losing numbers" needs possibly a bit explaining as to why a multiplication is needed (compare the above action when calculating the probabilities). —Preceding unsigned comment added by 87.202.31.226 (talk) 13:56, 26 July 2010 (UTC)

Also an explanation is needed as to why the ratio 1/Probability is needed. The canonical way to get the "odds" is (1-probability)/probability. —Preceding unsigned comment added by 87.202.31.226 (talk) 14:10, 26 July 2010 (UTC)

Possible link
Would be nice if http://lottery.merseyworld.com/Info/Chances.html could be linked in as it shows the working based on every ticket possible being purchased just makes it easier for people to understand i think — Preceding unsigned comment added by 86.29.180.115 (talk) 17:49, 13 October 2012 (UTC)

Probability to be squared?
I have often wondered if it would be more correct for lottery probabilities to be squared? In order to win a jackpot for example the player must select 6 numbers from 49 (without replacement) there is a 1 in 14 million chance of picking any particular combination of 6 numbers (although you are probably biased etc. let's say you borrowed a lottery ball machine to pick the numbers for your ticket). In order to win, the lottery ball machine then has to pick the same 6 numbers, on a separate occasion, also with a 1 in 14 million chance of picking a particular combination. Two independent random events that both have to happen to win the jackpot. P(A n B). so the chance of winning is actually (1/14 million)^2. Can someone who understands stats better than me tell me why this is wrong? — Preceding unsigned comment added by 138.253.94.111 (talk) 18:11, 20 March 2013 (UTC)


 * Heya! The way this works is that your ticket can effectively be ANY combination of 6 numbers from 49; it doesn't matter which combination you choose, but in order to win the jackpot you need to match that combination with the one drawn by the lottery machine. Another way to illustrate it is to think of the chance of rolling a double with two dice; it doesn't matter what the first die's value is, but the second one needs to match it. Therefore the probability of rolling a double is 1/6, not (1/6)^2. I hope that helps, feel free to ask for further clarificatino if you'd like :) JaeDyWolf ~ Baka-San (talk) 18:34, 20 March 2013 (UTC)


 * wow this really worked! - never thought I would see a reply, thank you. OK point taken - but isn't there something here about where and how the observation of the system is made? If I roll a dice twice there are 36 equally likely combinations that can result (21 if we ignore the order of the numbers though then not all with the same likelihood); 6 of these combinations are doubles, therefore P(double) = 6 /36 = 1/6, so I'm not sure where the matching bit comes into it? P(1 and 1) is still 1 in 36, as is P(2 and 1) or P(1 and 2) etc etc..
 * If I used a lotto machine to pick a combination of numbers for my ticket and then used the same machine in the real draw, surely the probability of the two selections giving the same numbers is (1/14 million)^2? (the calculation of the the 1/14 million probability has already accounted for being able to select the combination of numbers in any order I think). By extension swap in any means for random selection of numbers for picking the numbers for your ticket (quick dip, random number generator, kids birthdays) and the probabilities still hold? Thank you again for taking the time to discuss. — Preceding unsigned comment added by 138.253.94.111 (talk) 19:40, 20 March 2013 (UTC)


 * No trouble at all, I love to help when I can! With regards to the dice query, the "matching" part is another way of calculating the probability for certain events (such as rolling a double) and is useful for illustrative purposes but needn't be considered in your method. The reason I chose dice was because your question made me remember a |Yahtzee Numberphile video that demonstrates the same method but for five dice. At one minute into the video you'll see why the first die rolled affects calculation in the way that it does.
 * With regards to applying this to the lottery, I also had the same problem a while back. While it's true that the 6 numbers chosen out of 49 on your ticket had a ~1/14000000 chance of being chosen (assuming randomness for simplicity's sake), it doesn't affect the draw itself. If, instead, you correctly predicted a set of 6 numbers before you got your ticket, those numbers indeed appeared on your ticket and THEN won the jackpot, that sequence of events is does indeed have a probability of ~(1/14000000)^2. Yet another way of looking at it is that the probability of you having a ticket, given that you bought a ticket, is equal to 1. Hope this helps! You're free to ask me anything stats-related at any time :) JaeDyWolf ~ Baka-San (talk) 20:42, 20 March 2013 (UTC)


 * Aha! You said the magic word 'given'! Yes this makes sense to me now via the standard maths of conditional probability that leads to bayes theorem. P(lottery draws your numbers given you picked your numbers) equals P(lottery draws the combination) x P (you draw the combination) / P ( you draw the combination) = P (lottery draws the combination) = 1 in 14000000. Once you've bought the ticket, you have a 1 in 14 million chance. To an external observer of the human behaviour of playing and winning the lottery though, they might calculate the odds of a random human being winning the lotto as P(the person is willing to gamble) x P(random number generator picks a combination) x P(draw machine picks the same combination)? Something about this almost feels like quantum mechanics to me - you make an observation or selection and collapse the probability space, or maybe this is just nonsense. Many, many thanks again for talking this through with me.  — Preceding unsigned comment added by 2.126.79.101 (talk) 22:47, 20 March 2013 (UTC)
 * To understand odds or chances you best first play the game and study it.
 * If you play two pick 6's you are playing 6 to 12 numbers spread over 2 combinations. If you calculate for one game then your cannot have 2*6/6 with 12 numbers played.
 * Say that you if played 1 to 6 and 7 to 12. 7 to 12 is drawn and your results are: 0/6 and 6/6. You cannot have twice 6/6.
 * If you played two times 7 to 12, then you would have twice 6/6 but the jackpot generally is split through the amount of 6/6 winners.
 * On the long run playing 1 to 6 and 7 to 12, if all combinations were drawn once and you played them all, then yes.
 * If you play two pairs at keno where only 2/2 pays in a 20/70 then you play 2, 3 or 4 numbes in total. You can have 1 to 3 sets in your calculation. (...) 2A02:A03F:E03F:6300:187A:5CC2:AB92:A706 (talk) 14:49, 6 February 2023 (UTC)

Reciprocal of combination.
In: "An alternative method of calculating the odds is to never make the erroneous assumption that balls must be selected in a certain order. The odds of the first ball corresponding to one of the six chosen is 6/49; the odds of the second ball corresponding to one of the remaining five chosen is 5/48; and so on. This yields a final formula of...", the formula uses recipricals of the 6/49, 5/48, etc. This might be confusing to some. Maybe there is a way to bridge that. The combination number gives the number of possibilities but people want to know the odds, which is the reciprical of combination, right? Otherwise, what a great article, I wish textbooks were this clear. 71.139.178.219 (talk) 16:42, 19 May 2013 (UTC) 71.139.178.219 (talk) 16:45, 19 May 2013 (UTC)
 * I believe the article itself is incorrect; 6/49 is a probability and its equivalent odd would be 43 to 6 against (or other wording variations to that effect.) I've not heard of odds being referred to as the reciprocal of odds before and I don't think it's true for the general case, although I'd be willing to fold if somebody corrects me. I doubt I've fully answered yuor question but if you get back to me with a little clarification on what exactly is difficult then I'm more than happy to try again :) JaeDyWolf ~ Baka-San (talk) 16:33, 20 May 2013 (UTC)
 * I have since made a simple correction, calling those values probabilities, and I think I've seen where you're coming from. For the case of matching all six numbers, there is only one combination out of COMBIN(49,6) where all six match. That would make the probability of matching all six numbers the reciprocal of the number of combinations. Does that help? JaeDyWolf ~ Baka-San (talk) 16:43, 20 May 2013 (UTC)

Lead changes
I removed a large paragraph from the lead paragraph. With some copyediting, it could be put back in. Iamaplayer33 (talk) 20:53, 9 January 2017 (UTC)

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Odds of getting other possibilities
Hi, just a question or two regarding the general formula given in the section Odds of getting other possibilities in choosing 6 from 49

The given formula is:

$${K\choose B}{N-K\choose K-B}\over {N\choose K}$$, where $$N$$ is the number of balls in lottery, $$K$$ is the number of balls in a single ticket, and $$B$$ is the number of matching balls for a winning ticket.

However, applying this to a simple case where $$N = 4, K = 3, B = 1$$ results in an error:

$${3\choose 1}{1\choose 2}\over {4\choose 3}$$

with the problematic term being $${1\choose 2}$$ = $${1!}\over {-1! * 2!} $$ in the enumerator.

Many other seemingly valid choices e.g. $$N = 5, K = 4, B = 1$$ or $$N = 5, K = 4, B = 2$$ lead to similar errors.

Shouldn't the formula be valid for cases like these as well? If not, should there not be some form of restriction placed on permissible values (ideally with some explanation)?

Regards — Preceding unsigned comment added by Frikdt (talk • contribs) 19:00, 23 October 2017 (UTC)


 * The game usually is 6+1 from 49 balls while marking 6 numbers in one grid if it isn't a full system form where you can here put 7 to 14 crosses in one grid.
 * Think that you have 6 first drawn numbers, a 7 th drawn number and 49-6-1=42 not drawn numbers.
 * You have three subsets and let's call then n1, n2, n3 with k1, k2 and k3 for pick 6.
 * The total of the k's is 6. The total of the n's is 49.
 * Your formula for all cases to find the count of combinations is nCr on your calculator. The r is here our k.
 * So you enter: n1 nCr k1 * n2 nCr k2 * n3 nCr k3. Each k <= n and each k >=0.
 * You best do this on a spreadsheet. The total of your found combinations must be 49 nCr 6 or combin(49,6).
 * You notice that here there is only one 6/6, while in games like 20/70 there are many 6/6's.
 * In a 12/24 game the 0/12 and 12/12 are the jackpot. If you consider wheeling then your conditions there are <=4 or >= 8 for a minimum payout. 2A02:A03F:E03F:6300:2D88:F713:94F0:76BB (talk) 23:48, 6 February 2023 (UTC)

(deleting my own support for this to add the answer as I've figured it out)

Ok, so the problem with the scenario you've described, in a nutshell, is it's impossible to have it happen. This is the odds of getting EXACTLY B in common, not at least, and where both parties draw K numbers. I was super confused about this as well, but I've finally figured it out. The first scenario you describe is $$N = 4, K = 3, B = 1$$ which is a perfect example to demonstrate with. Basically that means there are 4 balls, 1-4. Now, if you and I both draw 3 balls from our own set of 1-4, what are the odds we have one in common?

0%. It's impossible for me to draw 3 numbers from 4, and you to as well, and us not have at least 2, if not all 3 in common, those are the only two scenarios. If I draw 1, 2, and 3, you can either draw a 4, and we'll have 2 in common, or not, and we'll have 3 in common. So if B is zero or one, the math breaks down because it's basically trying to find the odds of a scenario that doesn't exist.

If you punch in $$N = 4, K = 3, B = 2$$ you get 75%, and $$N = 4, K = 3, B = 3$$ gives you 25%, because that is all of the potentially valid results, so that's your 100%.

Hope this helps someone confused like me!

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