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Probability and Combinations

Determine the probabilities of events when arrangement order does not matter

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Beating the Odds

Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

Could you play all possible combinations of a lottery and guarantee a payout of more than what you spent? If you had a lot of money, this might be a great way to make even more money!

Sad But True

Consider the game Mega Millions. In each drawing, five balls are drawn from a set numbered 1 through 56, and one ball is drawn from a set numbered 1 through 46. The total number of possible combinations is about 176 million. If you buy one ticket that costs $1, you have a 1 in 176 million chance of winning the jackpot. If you had enough money, you could buy 176 million tickets with all the different combinations of numbers on them to ensure that you have a winning ticket.

Credit: Andrew Magill
Source: http://www.flickr.com/photos/amagill/3367543296/
License: CC BY-NC 3.0

There are times when the jackpot may grow beyond that 176 million mark. If it’s up at $300 million, would you be making a $124 million profit by buying all those combinations? Aside from the cash payout (the lottery actually awards much less than the advertised jackpot amount) and taxes (you have to pay Uncle Sam!), there′s also a chance that people besides you will get lucky and win as well. If that happened, you would have to split the jackpot with them, and you'd likely end up losing a lot of money. Say goodbye to that private island you were eyeing!

So while it is possible to make sure that you hit the jackpot, it′s not possible to guarantee that you will actually make a profit!

See for yourself: http://www.circlemud.org/jelson/megamillions/

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How can you use combinations to calculate the 176 million figure used above?

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Image Attributions

  1. [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0
  2. [2]^ Credit: Andrew Magill; Source: http://www.flickr.com/photos/amagill/3367543296/; License: CC BY-NC 3.0

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