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 LottoPoster Forums : INTRODUCTION AND GENERAL LOTTO RELATED DISCUSSION : Welcome and useful information on using the Forums. |
 Topic: UNDERSTANDING LOTTO ODDS AND SIMPLIFICATION |
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Colin F
Lotto Systems Tester Creator & Analyst 
To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678 |
   Topic: UNDERSTANDING LOTTO ODDS AND SIMPLIFICATIONPosted: May 27 2009 at 3:28pm |
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UNDERSTANDING LOTTO ODDS AND SIMPLIFICATION
by Colin Fairbrother Odds is a generic term which covers the different ways of expressing the chances of success for a given Lotto game and is best understood by adopting some flexibility towards the method of expression. Probability comes under the Odds umbrella and is the ratio of the chances of success for a given scenario with all the possibilities. Using the classic and very popular Lotto game where 6 integers (whole numbers or balls) are randomly chosen from 49 then the total possiblities are 13,983,816. You will find this game referred to as a 6/49 Lotto game with the forward slash used as a seperator and not to indicate division. The odds in favor for the 6/49 Lotto game are usually expressed as the ratio of playing one combination or line against all the possibilties ie 1 in 13,983,816 and 1 divided by 13,983,816 gives a probability of success for the unique winning number of 0.00000007 when one line or number is played. Some official Lotto sites will give Odds Against as in 13,983,816:1 or 13,983,816 to 1. Since 2003 I have posted on the internet bits and pieces on Odds in Lotto and in this article I will pull as much as I can together to aid understanding. Probably, the most misunderstood and likely to draw swift dissension concept is that of reduction or simplification ie 50/100 = 25/50 = 1/2. Playing 1 line your chances of getting 1st prize are 1 in 13,983,816. No problem there - every one agrees. However, if you play 2 lines your chances of getting 1st prize are 2 in 13,983,816 or 1 in 6,491,908. This reduction is not accepted by some as being valid. If we have 5 plays then 13,983,816 is not evenly divisible by 5 so it can't be simplified and we are stuck with 5 in 13,983,816. Playing 9 once again 13,983,816 can't be evenly divided by 9 but we can divide each side by 3 to get the simplified expression 3 in 4,661,272. Below is a table for the 6/49 Lotto game showing 192 cases where the odds may be expressed as 1 in something because 13,983,816 is evenly divisible by the number of plays. The concept becomes easier to understand when it is realized that this is just an average ratio dependent on adequate sampling where the difference is brought down to around the 5% mark compared with the theoretical probability calculation. If we enumerate the 13,983,816 combinations for a 6/49 Lotto game in lexicographical or numerical order producing an index at the same time then we would have both 6,991,908 odd and even indexes. ie
Index Combination Odd or Even Index
1 1 2 3 4 5 6 O
2 1 2 3 4 5 7 E
3 1 2 3 4 5 8 O
. . . . . . . .
. . . . . . . .
. . . . . . . .
13,983,816 44 45 46 47 48 49 E
Now, if you take all the 6/49 Lotto games around the world to date and put them into a table while still retaining the draw date order you will have some 19,577 lines with 24 repeats and a difference between the odds and even indexes from the full set as described previously of around 5% which is considered normal. Currently, it favours even indexes but over the next 10 to 20 years it could drift the other way. Using our odds and even indexes the single winning number has either an odd or an even index - it can't have both. Allocating 1 ticket to the odd indexes and the other to the even we see the valid ratio is 1 in 6,991,908. But as previously explained we could have a run of 13 even or odd indexes for the draw numbers so this ratio must be considered as valid only within the context of adequate sampling. Playing just 1 line or ticket there is no guarantee your line will come up in 13,983,816 draws and in fact for that number of draws there will be repeats with only close to 63% unique. Playing 2 lines or tickets, maybe, no guarantee and how relevant is 140,000 years? Checking out the various 6/49 games you will find even though you are theoretically playing half the possibilities you can still have a run of 13 odd or even indexes for 19,577 draws (see table below), Therefore, an adequate sample would have to be at least around 100 draws. In other words valuing 1 ticket at $1 and 1st prize at $5,000,000 you spend $600,491,908 to get $250,000,000 plus some secondary prizes. In the USA you are taxed on winnings so you can half that amount. Your stake for such a foolhardy endeavour would have to be enormous as at times you could lose consecutively some $80,000,000 all efforts progressing towards a loss of some $350,000,000 - good for the Lotto operators bad for anyone contemplating the idea of deliberately setting out to get a 1st prize win with an unlimited budget. RunCnt Cnt 13 1 12 3 11 5 10 9 9 20 8 39 7 62 6 163 5 278 4 640 3 1237 2 2489 1 4838 Odds for winning 1st Prize in 6/49 Lotto game as plays per draw are increased.
Colin Fairbrother |
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