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The Odds of a Lottery
| Calculation explained in choosing 6 from 49 |
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 In a typical 6/49 lotto, 6 numbers are drawn from a range of 49 and if the 6 numbers on a ticket match the numbers drawn, the ticket holder is a jackpot winner—this is true no matter the order in which the numbers appear. The probability of this happening is 1 in 14 million (13,983,816 to be exact). |
 The relatively small chance of winning can be demonstrated as follows:
Starting with a bag of 49 differently-numbered lottery balls, there is clearly a 1 in 49 chance of predicting the number of the first ball selected from 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. |
 Thus, each of the 49 ways of choosing the first number has 48 different ways of choosing the second. This means that the odds of correctly predicting 2 numbers drawn from 49 is calculated as 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but of course someone picking numbers would have gotten to this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49 is calculated as 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44. This works out to a very large number, 10,068,347,520, which is however much bigger than the 14 million stated above. |
 The last step is to understand that the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any set of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 720 ways they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as 49! / (6! × (49 - 6)!) |
 In most popular spreadsheets, this function, called the combination function is denoted COMBIN(n, k). For example, COMBIN(49, 6) (the calculation shown above), would return 13,983,816. |
 Taken as a class, the number of possible combinations for a given lottery can be referred to as the "number space." "Coverage" is the percentage of a lottery's number space that is in play for a given drawing. With the Washington State Lottery, the coverage for the record lottery of 2007-03-06 was 70%. 7 out of every 10 possible number combinations had been chosen for this lottery. |
 The 4th Market lottery seems to be an exception to all other types of lotteries given that the outcome of all the stock positions could in theory be predicted, If we start with a pure random selection then the odds of getting the share position correct for each of the stock market shares are 10 Factorial (10x9x8x7x6x5x4x3x2x1) = 3,628,800 - much lower odds than any other state run lottery. Now we factor in knowledge of the shares in the stock market and the odds could dramatically change in the players favour. Calculating the skill factor is not possible because there are too many variables, but in theory a good day trader could cut those odds in half. |
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