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probability types

Topic closed. 18 replies. Last post 10 years ago by guesser.

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jordi marey's avatar - Lottery-039.jpg

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Posted: September 21, 2006, 6:07 am - IP Logged

 

John Schofield's article in the Personal Finance section contributes a number of calculations to Maclean's Magazine. The expectations were based on the payoffs of the September 7, 1996, $10,000,000 lottery. At the time the amount was quite unusual.  He used the amount $2,217,321 which was the jackpot a week earlier, and which is a more typical figure.  Provided below are some useful information concerning Lotto, both probability calculations and strategies. 

Probability Background

The theory of probability is a mathematical model for understanding, describing, analyzing, and predicting phenomena which exhibit random fluctuations in their behavior. These are characterized by the property that small scale prediction cannot be achieved with high accuracy, but there is remarkable long-run regularity which can be studied mathematically. This subject has roots in games of chance (in particular in a correspondence between the philosopher Blaise Pascal and a French nobleman the Chevalier de Mere) but has developed far beyond to address fundamental issues in genetics, physics, medicine, communications theory, and software reliability, for instance.

There are three types of probabilities, differing only in interpretation but subject to the same rules:

Mathematical

The theoretical development based on equally likely outcomes and their far-reaching generalization into measure theory.

Empirical

Events are assigned probabilities based on empirical observations from the past.

Subjective

Championed by the Italian mathematician Bruno de Finetti and views probabilities as personal reflections of an individual's opinion about an event.

The probabilities associated with Lotto 6/49 are determined by calculations based on equally likely outcomes but their interpretation is empirical, as follows. If an experiment (such as a weekly Lotto drawing) is repeated under similar conditions many times (mathematically the requisite number of times must approach infinity) then the probability of an event is the long run proportion of experimental repetitions on which the event occurs. (This is called a law of large numbers. It is a mathematical result only.) As illustration, the probability of obtaining a fourth place prize is 0.0009686. This means that in 26,000 Lotto draws, each particular choice of six numbers will yield a fourth place prize (matching exactly 4 numbers) on approximately 0.0009686 of those plays, that is 26000 x 0.000986 = 25 times. There is no guarantee that this will happen 25 times. The actual number of times is random and subject to what is called the Poisson approximation. The figure of 25 represents an expected number of winnings and is a useful measure by which to compare the performance of different strategies.

How the Probabilities are Calculated

A brief description of the combinatorics needed here to grasp the Lotto 6/49 probabilities.

Jackpot (all six winning numbers selected)

There are a total of 13,983,816 different groups of six numbers which could be drawn from the set {1, 2, ... , 49}. To see this we observe that there are 49 possibilities for the first number drawn, following which there are 48 possibilities for the second number, 47 for the third, 46 for the fourth, 45 for the fifth, and 44 for the sixth. If we multiply the numbers 49 x 48 x 47 x 46 x 45 x 44 we get 10,068,347,520. However, each possible group of six numbers (combination) can be drawn in different ways depending on which number in the group was drawn first, which was drawn second, and so on. There are 6 choices for the first, 5 for the second, 4 for the third, 3 for the fourth, 2 for the fifth, and 1 for the sixth. Multiply these numbers out to arrive at 6 x 5 x 4 x 3 x 2 x 1 = 720. We then need to divide 10,068,347,520 by 720 to arrive at the figure 13,983,816 as the number of different groups of six numbers (different picks). Since all numbers are assumed to be equally likely and since the probability of some number being drawn must be one, it follows that each pick of six numbers has a probability of 1/13,983,816 = 0.00000007151. This is roughly the same probability as obtaining 24 heads in succession when flipping a fair coin!

Second Prize (five winning numbers + bonus)

The pick of six must include 5 winning numbers plus the bonus. Since 5 of the six winning numbers must be picked, this means that one of the winning numbers must be excluded. There are six possibilities for the choice of excluded number and hence there are six ways for a pick of six to win the second place prize. The probability is thus 6/13,983,816 = 0.0000004291 which translates into odds against of 2,330,635:1.

Third Prize (five winning numbers selected, bonus number not selected)

As in the second prize there are six ways for a pick of six to include exactly five of the six drawn numbers. The remaining number must be one of the 42 numbers left over after the six winning numbers and the bonus number have been excluded. Thus there are a total of 6 x 42 = 252 ways for a pick of six to win the third prize. This becomes a probability of 252/13,983,816 = 0.00001802 or, equivalently, odds against of 55490.3:1.

Fourth Prize (four winning numbers selected)

There are 15 ways to include four of the six winning numbers and 903 ways to include two of the 43 non-winning numbers for a total of 15 x 903 = 13,545 ways for a pick of six to win the third prize, which works out to a probability of 13,545/13,983,816 = 0.0009686, that is odds against of 1031.4:1.

Fifth Prize (three winning numbers selected)

There are 20 ways to include three of the six winning numbers and 12,341 ways to include three of the 43 non-winning numbers for a total of 20 x 12,341 = 246,820 ways for a pick of six to win the fourth prize, which works out to a probability of 246,820/13,983,816 = 0.01765, that is odds against of 55.7:1.

 

    jordi marey's avatar - Lottery-039.jpg

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    Posted: September 21, 2006, 6:09 am - IP Logged

    empirical seems the most useful for the daily Picks.

      Thoth's avatar - binary
      Findlay, Ohio
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      Posted: September 21, 2006, 12:26 pm - IP Logged

      empirical seems the most useful for the daily Picks.

      Interesting post Jordi

      For the daily games (pick 3/pick 4) you can actually calculate all of the true probabilities using the methods that are outlined by combinatorics.  Examining empirical amounts of data (large histories of previous results that is) will verify the accuracy of the probability calculated by the combinatorics:  if something has a 20% chance of occuring, it will occur 20% of the time, if something has only a 1% chance of occuring, it will only occur about 1% of the time, etc.

      Finding a a real mathematical advantage to the games will probably entail finding an event that constantly performes at a much higher percentage than that of its true probability.

      ~Probability=Odds in Motion~

        RJOh's avatar - chipmunk
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        Posted: September 21, 2006, 1:18 pm - IP Logged

        Thoth,

        Are you suggesting that the significant of a particular event repeating is more important than the repeatable of particular numbers?  For example (the event)- 26% of PowerBall drawings have no numbers from the previous three drawings which means by eliminating the 12-15 numbers in the previous three drawings a player could increase his odds of winning the jackpot or second prize 400-500% every time that event happened, but would having the better odds 26% of the time and no chance 74% of the time be better then having a chance all the time with the normal odds?

         * you don't need to buy more tickets, just buy a winning ticket * 
           
                     Evil Looking       

          guesser's avatar - Lottery-017.jpg

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          Posted: September 21, 2006, 4:15 pm - IP Logged

          You are on the right track, but you need to look further at the 'age' of a number, and what other numbers fall into that 'age' category.

            guesser's avatar - Lottery-017.jpg

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            Posted: September 21, 2006, 4:46 pm - IP Logged

            Danged system threw me out. maybe it's an Omen.....

             

            Anyway, RJoH, your stats are flawed, I generally only go back 20 games because to go back further than than in generalities is misleading. Trends and tendencies change.

             

            Of the last 20 games, 17 have had repeats from the prior 3 games, going back 40 games and it's 34/40, go back 60 and it's 47/60.

            This makes sense because this is where a majority of the 55 numbers are (top 3 age groups), it's the things that don't make sense that trick us, this is where the true randomness comes into play.   I picked #7 to repeat from the last game, and while I did not pick #8 to repeat, when you look at it, it's not a surprise.   #7 NOT hitting would have surprised me more than #8 hitting did. 

              RJOh's avatar - chipmunk
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              Posted: September 21, 2006, 6:30 pm - IP Logged

              The above observations about 26% of the time was just one example.  Going the other way if one looks at the previous 16 drawings which usually covers 38-46 numbers, all the winning numbers are in that group 29% of the time and if you only include numbers that have hit 1-2 times, the pool is reduced to 26-40 numbers and have all the winning numbers 7% of the time.

              As Thoth suggests, the trick is to find an event that has a mathematical advantage by the number of times it occurs.  I suspect the observations I've made are normal for numbers picked from a pool 55 numbers.

               * you don't need to buy more tickets, just buy a winning ticket * 
                 
                           Evil Looking       

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                NASHVILLE, TENN
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                Posted: September 21, 2006, 9:02 pm - IP Logged

                Thoth,

                Are you suggesting that the significant of a particular event repeating is more important than the repeatable of particular numbers?  For example (the event)- 26% of PowerBall drawings have no numbers from the previous three drawings which means by eliminating the 12-15 numbers in the previous three drawings a player could increase his odds of winning the jackpot or second prize 400-500% every time that event happened, but would having the better odds 26% of the time and no chance 74% of the time be better then having a chance all the time with the normal odds?

                RJOH

                IMHO, I would settle for having a 1% chance if I knew that 1% would result in a win.  Winning PB once a year is, to me, an attainable goal.  So what if I lose 99% of the time?  If you do it right, once is enough.

                  guesser's avatar - Lottery-017.jpg

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                  Posted: September 21, 2006, 9:26 pm - IP Logged

                  The above observations about 26% of the time was just one example.  Going the other way if one looks at the previous 16 drawings which usually covers 38-46 numbers, all the winning numbers are in that group 29% of the time and if you only include numbers that have hit 1-2 times, the pool is reduced to 26-40 numbers and have all the winning numbers 7% of the time.

                  As Thoth suggests, the trick is to find an event that has a mathematical advantage by the number of times it occurs.  I suspect the observations I've made are normal for numbers picked from a pool 55 numbers.

                  I may be wrong, but according to what you are writing, I think you are selling yourself short if you only look at the numbers comprising the last 16 games.  But your hypothesis is correct - but only 33% of the time.  You may as well just play games every third drawing.

                  You and me look at the 'numbers' totally differently, and I'm absolutely not going to tell you your way is 'wrong', but I understand how my way works, maybe one of these days one of us - or both of us - will get lucky.

                  Everyone has a 'system', but if we hit a jackpot, will it be because of a 'system', or will it be from luck ?  

                    Thoth's avatar - binary
                    Findlay, Ohio
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                    Posted: September 21, 2006, 9:46 pm - IP Logged

                    Thoth,

                    Are you suggesting that the significant of a particular event repeating is more important than the repeatable of particular numbers?  For example (the event)- 26% of PowerBall drawings have no numbers from the previous three drawings which means by eliminating the 12-15 numbers in the previous three drawings a player could increase his odds of winning the jackpot or second prize 400-500% every time that event happened, but would having the better odds 26% of the time and no chance 74% of the time be better then having a chance all the time with the normal odds?

                    Honestly I don't play PB (not available in Ohio) and I rarely play the Mega Millions, so I dont know the particulars of the strategy you mentioned.  But, if I was playing PB with the choices you laid out, I would probably try to anticipate the 26% and choose my numbers from that group.  On the other hand, playing or expecting digits to repeat from X amount of games back is also a good stategy.

                    What I meant in my post was that if any real mathematical advantage is ever found for winning the games that it will probably have to be something that inexplicitly occurs against the known laws of probability.  And I don't mean particular digits or combinations which have a statistically higher frequency than others, because the probability of everything hitting equally is astronomically small. 

                    What I'm imaging is an event that has a specific and constant probability, but at the same time performs at a much higher percentage than what it has mathemtically been alotted according to its probability.  As an inflated example: say an event has a probability of .05 or 5%. This event should only occur 5% of the time over the long term.  Now imagine that the event actually occurs 10% to 15% of the time—not just in one state, but in every state that has the same game.  Of course finding such an event will most likely never happen - if its even possible.

                    ~Probability=Odds in Motion~

                      guesser's avatar - Lottery-017.jpg

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                      Posted: September 21, 2006, 11:04 pm - IP Logged

                      If you play only numbers that are 16 games old or newer, then you only have a chance in 'about' 33% of the games.

                       

                      It comes back to knowing when to play what, which is dependent on how the recent games have hit.

                       

                      If you pick 4 out of 5 numbers from the prior 16 games, then you stand a 66% chance over three games, it's that 5th number that's killer.

                       

                      I don't exactly wet my pants when I get 3 x 5, I'm aiming for 5 x 5, but not everybody is like me. I aim for a 5 x 5, and I never count on a jackpot, but if it happens, great.

                        RJOh's avatar - chipmunk
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                        Posted: September 22, 2006, 9:08 am - IP Logged

                        Using the 16 games in the last example or using the 3 games to eliminate numbers in  the first example was just a way to define a pool of numbers less than 55 that had done well in the past.  Other parameters could probably do even better.  If I could reduce the pool to 10 numbers that had in the past hit 5% of the time then I would just play those numbers and wait for the event to happen.

                        In the MM and PB challenges in the jackpot section, participants have been picking 15 white balls and 5 bonus balls for every drawings and no one has had all 5+1 winning numbers yet. 

                         * you don't need to buy more tickets, just buy a winning ticket * 
                           
                                     Evil Looking       

                          guesser's avatar - Lottery-017.jpg

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                          Posted: September 22, 2006, 3:46 pm - IP Logged

                          You and I think alike in that we choose numbers based on an event likely to happen in the near future, but how we choose our pool of numbers is quite different.  I don't 'chase' numbers, I chase events/trends/scenarios, hopefully I will hit one  one of these days.

                            RJOh's avatar - chipmunk
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                            Posted: September 22, 2006, 4:42 pm - IP Logged

                            guesser,

                            I agree we are doing somewhat similar things.  I don't really chase numbers because I define the events using letters and in my programs I don't compare numbers I compare charcters for example I don't use 1=1 but "01"="01", the "01" could easily be "AB"  or "Bb" the way my program works.  In the final step I convert the letter characters to number characters based on the most recent drawings so the characters " B" or " k" could be any number.  Good luck to you, I'm still trying to find that unique event to cover. 

                             * you don't need to buy more tickets, just buy a winning ticket * 
                               
                                         Evil Looking       

                              guesser's avatar - Lottery-017.jpg

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                              Posted: September 23, 2006, 12:24 am - IP Logged

                              Using the 16 games in the last example or using the 3 games to eliminate numbers in  the first example was just a way to define a pool of numbers less than 55 that had done well in the past.  Other parameters could probably do even better.  If I could reduce the pool to 10 numbers that had in the past hit 5% of the time then I would just play those numbers and wait for the event to happen.

                              In the MM and PB challenges in the jackpot section, participants have been picking 15 white balls and 5 bonus balls for every drawings and no one has had all 5+1 winning numbers yet. 

                              It's almost like you are choosing numbers based on 'hot/cold' based on you saying 'had done well in the past'.

                               

                              There are some numbers that DO follow a hot/cold play, but I don't think you can predict with any certainty when a number will turn hot or cold, or for how long. Or which numbers.   I see numbers that hit 3 out of 6 games, then don't hit again for 30-40 games, and I see numbers that hit, skip 8 games, hit, skip 10 games, hit, skip 8 games, so you think you can time it ? Nope - it may hit again in 2 games, then die for 50 games.  I see some, most, with no pattern whatsoever.

                              As I said before, look at us here - we keep our our stats, and we have a 'Powerball Challenge' where we pick 15 numbers, and so far, I think one guy has picked 4 x 5 one time, or maybe it was 5 x 5, but he didn't have them all in the same game.  That's a pretty good indication that all the number chasing and stat keeping we do will never account for the 2 numbers that always hit from nowhere.

                              So, I pick my 3 'good' numbers, and then I pull 2 numbers from nowhere, and ya know what ?  

                              I'm no better off than anyone else, but I usually get 2 of my 'good' numbers to hit, and more often than not, one of the numbers I pick from nowhere hits.