Calculations for 1 in 20 Chances of Winning
How do we work out that the chances of winning are 1 in 20 if there are 3,129 draws out of 61,000 tickets?
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A maximum of 61,000 tickets will be sold and will be eligible to be drawn. |
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There are 3,131 prizes overall and one prize is awarded to each ticket drawn. |
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One ticket number is drawn at a time. After each draw, the winning ticket is recorded and is then re-entered into the drawing process. |
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Two of the draws will be a Bonus Draw and an Early Bird draw which are available only to tickets purchased prior to certain dates. |
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Each ticket number is equally eligible in all draws, regardless of whether it has already won one or more prizes, subject to the Bonus and Early Bird draws only being available for tickets purchased prior to certain dates. |
When looking at the chances of winning, it is important to remember that this is only an expected outcome, based on probability calculations. These calculations can be undertaken by a mathematician or actuary, or by a person who understands probability calculations.
After the draw for the Lottery has occurred, if all of the prize winning tickets are counted, and then compared with the total number of tickets that were sold, the proportion of prize winning tickets is expected to be greater than 1 in 20. This means that we expect that there should be at least 3,050 prize winning tickets (which is 61,000 divided by 20).
Since a winning number is returned to the draw, it is possible for a ticket to win more than one prize.
If we look at what happens after the first prize draw, there will be 60,999 tickets that did not win a prize, and only one that has won a prize. For each draw, there will always be 60,999 tickets that did not win a prize at that draw.
After 100 prize draws, what is the chance that a ticket has not won any prize at all? The chance is calculated as (60,999/61,000) multiplied by itself 100 times, which the mathematicians call "raised to the power of 100", If we multiply (60,999 divided by 61,000) by itself 100 times, we get 0.998362 (to 6 decimals). If the chance of a ticket not winning a single prize is 0.998362, this means that the chance of a ticket winning a prize (that is at least one prize) after 100 draws is the difference between that and 1.0, which is 0.001638.
After 1,000 draws, the chance of a ticket not winning any prize is calculated as (60,999 divided by 61,000) multiplied by itself 1,000 times, which is 0.983740. This means that the chance of a ticket winning a prize (that is at least one prize) after 1,000 draws is the difference between that and 1.0, which is 0.016260.
After 3,129 draws, the chance of a ticket not winning any prize is calculated as (60,999 divided by 61,000) multiplied by itself 3,129 times, which is 0.949997. This means that the chance of a ticket winning a prize (that is at least one prize) after 3,129 draws is the difference between that and 1.0, which is 0.050002. This can be expressed as a 1 in 19.999 chance, which is a little better than a 1 in 20 chance of a ticket winning a prize.
Our sincerest thanks to the following qualified professionals who have confirmed the accuracy of the calculation and the methodology to determine chances of winning.
Peter Crump,
Fellow of the Institute of Actuaries of Australia
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