Calculate the odd of winning Powerball in R

By Xianjun Dong

This Wednesday’s Powerball grand prize already climbed up to $1.5 BILLION. If you choose to cash out, it would be $930 million. And it keeps increasing…
So, what’s the odd of winning the jackpot prize?
Here is the game rule according to Powerball.com:

…we draw five white balls out of a drum with 69 balls and one red ball out of a drum with 26 red balls.

We can calculate the total number of different combinations in R:

> choose(69,5)*26
[1] 292201338

If we are super super lucky to win the Jackpot of $930 million cash value, given that we have to pay 39.6% as federal tax, how much we expect to return for a $2 investment? (Of course, everyone expect to win the $1.5 billion jackpot)

> 930e9*(1-0.396)/(choose(69,5)*26)
[1] 1922.373

This is a pretty good investment, comparing to the $40 million last week. But we know it’s not 100% guaranteed.
If we want to be 100% guaranteed, we have to buy all 292 million combinations. In that case, can we earn?

> choose(69,5)*26*2
[1] 584,402,676

> 930000000 + 1000000*choose(25,1) + 50000*choose(69-5,1) + 100*choose(5,4)*choose(69-5,1)*choose(25,1) + 100*choose(5,3)*choose(69-5,2) + 7*choose(5,3)*choose(69-5,2)*choose(25,1) + 7*choose(5,2)*choose(69-5,3) + 4*choose(5,1)*choose(69-5,4) + 4*choose(69-5,5)
[1] 1,010,666,048

> 1010666048 * (1-0.396)
[1] 610,442,293

That’s still more than what we paid. Why don’t we do that?
Remember, people share the prize if multiple persons got the same winning number, which we don’t know. Image may be NSFW.
Clik here to view.
Just some fun!
Image may be NSFW.
Clik here to view.

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References:
1. Fascinating Math Behind Why You Won’t Win Powerball (http://www.wired.com/2016/01/the-fascinating-math-behind-why-you-wont-win-powerball/)
2. Tax for lottery (http://classroom.synonym.com/much-federal-taxes-held-lottery-winnings-20644.html)

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