# Calculate the odd of winning Powerball in R

January 12, 2016
By

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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)

``> 930e6*(1-0.396)/(choose(69,5)*26)[1] 1.92``

Actually it’s not a good investment. (Thanks for the comment below. I made a mistake; 930million should be 930e6, not 930e9).
If we want to be 100% guaranteed, we have to buy all 292 million combinations. In that case, can we earn?

``# this is what we pay> choose(69,5)*26*2[1] 584,402,676# this is what we earn in total, before tax> 930000000 + 1000000*choose(25,1) + 50000*choose(5,4)*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] 1,023,466,048# Nearly \$5 billion!!! Then we need to pay 40% of tax. Maybe not for the minor prize, let's simplify it for all.> 1023466048 * (1-0.396)[1] 618,173,493``

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. 😀

Just some fun! 🙂

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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