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Odds of winning lotto


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If one man bought $20 worth of lotto tickets a week for a year and another man bought $1040 once a year which man would be more likely to win the lotto?
Assuming he doesn’t do something silly like buy all of his tickets with the same number, the man who buys many tickets for a single drawing is more likely to win than the many who buys the same number of tickets divided among many drawings.

 

Specifically, if the probability of winning a single drawing with $1040 worth of tickets is [math]p[/math], the probability of winning one or more of 52 drawings with $20 worth of tickets is [math]1 - \left(1 - \frac{p}{52} \right)^{52}[/math].

 

If [math]p[/math] is very small, as it is for most lotteries, these probabilities are nearly the same. For example, if [math]p = \frac{1}{1000000}[/math], [math]1 - \left(1 - \frac{p}{52} \right)^{52} \dot=\frac{1}{1000000.49}[/math].

 

Taken to the extreme, if the man saved his money and bought one of all of the possible tickets for a single drawing, his probability of winning would be 1. If instead he used the same money to buy 1/52 of the possible numbers, in 52 drawings, his probability of at least one win would be about 0.6325.

 

On at least one occasion, folk have used this trick to spend a few million to win a multi-million lottery payoff. After it was done once in Virginia, USA, winning a $1,300,000/year for 20 year payoff with $5,000,000 worth of $1 tickets (due to technical difficultys, the group failed to cover all roughtly 7,000,000 numbers), a law was passed banning it. (see Group Invests $5 Million To Hedge Bets in Lottery - New York Times)

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Lotto (Monday, Wednesday & Saturday)

Division Required to Win in A Single Game Panel

Odds based on 1 Game

:: NSW Lotteries ::

 

All 6 winning numbers 8,145,060 : 1

 

I would guess--not being a mathematician,-- investing a year's purchases would give you better odds.

Then, as my favorite author--Terry Pratchett says "Mullion to one chances happen every day" (?--or similar).

 

On all on line lottery sites in NSW odds are given by law, as well as links to Gambler's Anonymous.

:: NSW Lotteries ::

 

I think Marx should of said "Lotteries are the opium of the poor"

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There are other tricks to the lottery. If the jackpot is bigger than the odds then you will eventually win and profit, although it may take millenia to realize it. The Mega Millions in the US starts with a $12/m jackpot. It is drawn on Tuesday and Friday. Each time there is no winner the jackpot goes up. The odds of hitting the jackpot on any given drawing by playing one number is 175,711,536:1. So if you only play when the jackpot is higher than that then you are an investor, not a gambler. :)

 

If you buy multiple numbers then you can increase your odds, and reduce what the payoff needs to be to eventually profit.

 

The bugaboo in this is splitting the pot. If any number people hit the same number the jackpot is divided among the winning tickets. So one of the tricks is to pick numbers that are less likely to be picked by other people. Many people for instance play birthdays of family members, so when the winning numbers are between 1 and 31 there are far more shared jackpots than when some of the numbers are above 31. Avoid numbers of popular athletes, like 23. If you can think about human tendencies you will figure out what numbers to avoid to keep yourself from picking winning numbers that will force you to share the jackpot.

 

Bill

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Lotto (Monday, Wednesday & Saturday)

Division Required to Win in A Single Game Panel

Odds based on 1 Game

:: NSW Lotteries ::

 

All 6 winning numbers 8,145,060 : 1

Simple drawings like these are good opportunities to use simple probability.

 

The base odds of winning are just a simple selection of 6 of 45 balls, order not important:

[math]\frac{6!(45-6)!}{45!} = \frac{6\cdot5\cdot4\cdot3\cdot2}{45\cdot44\cdot43\cdot42\cdot41\cdot40} = \frac{1}{8145060}[/math]

 

Notice that the link gives not only the odds based on 1 game (8145060:1), but the odds based on 18 games (452503:1).

 

This is also just simple probability:

[math]1-\left(1-\frac{1}{8145060}\right)^{18} \dot = \frac{1}{452504}[/math]

 

If, instead of buying 18 cards, you bought 1000, your odds would be approximately:

[math]1-\left(1-\frac{1}{8145060}\right)^{1000} \dot = \frac{1}{8146}[/math]

 

If you bought 8145060 cards (carefully avoiding 2 with the same numbers, and assuming this is legal) you’d be sure to win – though you might win less than you spent, and even if you won more, it might take years to reach the break-even point, if it’s awarded in yearly installments.

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The Mega Millions in the US starts with a $12/m jackpot. It is drawn on Tuesday and Friday. Each time there is no winner the jackpot goes up. The odds of hitting the jackpot on any given drawing by playing one number is 175,711,536:1.

 

Large numbers are frequently hard to internalize, at least for me. Some politicians like to use the example of how many dollar bills it would take to get to the moon and back. I propose a more direct example.

 

According to this site: Fatality Analysis Reporting System

 

In 2008, there were about 3.402e4 fatal auto accidents in the US. (Number of accidents, not number of deaths). There were also a total of 2.926e12 miles driven. If you take some liberties in the statistical analysis and assume the place you purchase the lotto ticket from is only one mile away, and all fatal accidents are equally likely to happen during every mile driven, then you have a 1:8.601e7 chance of being involved in a fatal accident while driving to get your ticket. Or stated another way, the odds of you being involved in a fatal wreck on the way to purchase your ticket is 86,010,000:1. Compare this to the odds of the mega millions listed by TheBigDog.

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simple drawings like these are good opportunities to use simple probability.

 

The base odds of winning are just a simple selection of 6 of 45 balls, order not important:

[math]\frac{6!(45-6)!}{45!} = \frac{6\cdot5\cdot4\cdot3\cdot2}{45\cdot44\cdot43\cdot42\cdot41\cdot40} = \frac{1}{8145060}[/math]

 

notice that the link gives not only the odds based on 1 game (8145060:1), but the odds based on 18 games (452503:1).

 

Isn't the factorial of 45 ending on one?

 

45*44*43...*2*1

 

:confused:

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