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Lady of the lake, version #938

Saturday, May 18, 2013

Powerballin'

I was a bit astounded to hear that out of the millions of tickets sold for the current $600,000,000.00 Powerball jackpot, only 80% of the possible number combinations have been selected. I decided to look into the statistics a bit. Here is a link to  a page illustrating the frequency that balls are selected, both as white balls and power ball.

56, 26 and 23 are the numbers with the historically highest chance of being picked as a white ball, with the corresponding frequency of 18, 17 and 19. 29 is by far the winning number as a powerball selection.

Your odds of winning the Powerball this week, about 1 in 175.2 million. But you have to buy a ticket.

I read a book once on the science of picking lottery numbers and the author stated that your best chances are picking the upper strata of numbers and mixing in some numbers that are close to each other sequentially. It only stands to reason that you want to pick the numbers that most people don't pick.

Here are a couple links to sites regarding power ball numerical statistics. Business Insider and Daily Herald. And a good wikipedia article on lottery mathematics.
In a typical 6/49 game, six numbers are drawn from a range of 49 and if the six numbers on a ticket match the numbers drawn, the ticket holder is a jackpot winner—this is true no matter in which order the numbers appear. The probability of this happening is 1 in 13,983,816.
This small chance of winning can be demonstrated as follows:
Starting with a bag of 49 differently-numbered lottery balls, there are 49 different but equally likely ways of choosing the number of the first ball selected from the bag, and so there is a 1 in 49 chance of predicting the number correctly. 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 for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but of course we could have gotten to this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49, again in the correct order, is 1 in 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as . This works out to a very large number, 10,068,347,520, which is 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 orders in which they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816...

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