Lotto Probability

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Lee Kelly
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Lotto Probability

Post by Lee Kelly »

My local football club runs a weekly lotto consisting of 4 numbers and bonus ball between 1-28.. after the four numbers are drawn they are re-entered into the pot were the bonus is drawn, e.g 3,10,19,27 b10 the current jackpot is 13500 and has been for over 4 years, i am wondering what is the probability that someone could achieve the jackpot as its the bonus that has been the problem it has never went!!
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Michael Wallace
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Re: Lotto Probability

Post by Michael Wallace »

There are 20,475 ways of drawing four numbers from 28, and so if you then have to match the bonus as well (at odds of 1 in 28) that gives you a 1 in 573,300 of winning. Assuming I've understood the question correctly, of course.
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Innis Carson
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Re: Lotto Probability

Post by Innis Carson »

Do you choose 4 normal numbers, and then another one which is designated as the bonus ball? Or just 5 interchangeable numbers?
Lee Kelly
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Re: Lotto Probability

Post by Lee Kelly »

yeah 4 normal first innis from 1-28 then a bonus seperate but the bonus is drawn after with all 28 balls back in so it can be the same as one of the first 4 numbers sorry misread the question yeah thats the way it is on the slip (bonus is highlighted) i normally pick a different one from the four but i have seen it be the same
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Innis Carson
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Re: Lotto Probability

Post by Innis Carson »

In that case yeah, pretty sure Michael's right. So you should expect a jackpot win once every 11,000 years or so. Not too surprising that it's lasted 4!
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Steve Balog
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Re: Lotto Probability

Post by Steve Balog »

To expand a bit on Michael's post:

The first part of the draw, picking four numbers without replacement from a pool of 28, is called a combination. It can be written as 28C4, or the number of ways to choose 4 numbers from 28 where order doesn't matter. This is equal to 28!/((24!)(4!)), where n! is the product of all integers from 1 to n. This reduces to 28*27*26*25/(4*3*2*1), which is, as Michael said, 20,475.

Even ignoring the bonus ball, ]the expected value of the time required to just match the first four numbers is 393.75/(the number of people who play a week) years.

Now, once that once-in-a-lifetime event happens, there's only a 1/28 chance that it will also match the bonus ball. Now we have the lottery being won, on average, about once in a person's life if 138 people are playing it a week (assume a lifespan of 80 years). But that's just an average - there's about a e^-1 = 36.79% chance that the lottery will not be won in a person's life.

The only reason lotteries that have more numbers in them are won more often than once every four years is that they have a ton more people playing and thus have more opportunities for that infinitesimal chance of a win happening. If 10,000 people played your lottery a week, you'd get a winner almost once a year, about. If you give me about how many people play in your lottery a week I can tell you how long about you can expect to wait for a win, what the odds of it not being won in the 4 years it's been up are, etc.

PS: It doesn't matter at all if you pick a different number for the bonus for the four than your first four picks, or one of the same numbers. The probability of every ball being picked as the bonus, no matter if it was one of the first four picked or not, is 1/28.
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