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        riddles >> medium >> Powerball Expected Value
        
(Message started by: marsh8472 on Nov 28^th, 2012, 12:13am)

Title: Powerball Expected Value
Post by marsh8472 on Nov 28^th, 2012, 12:13am

What is the expected return of a powerball ticket.

Assumptions made for this problem

1) ticket costs $2
2) J = jackpot value
3) T = total people playing
4) Each person's ticket numbers are chosen at random
5) 5 balls selected without replacement from 59 balls labled 1-59 and a powerball selected from 35 possible balls labeled 1-35
6) Jackpots are divided evenly among the winners
7) powerball prizes are listed here http://en.wikipedia.org/wiki/Powerball or at www.powerball.com
8) assume no one's playing powerplay

I think the trickiest part of this is accounting for every possible outcome of jackpot winners. It's theoretically possible that everyone could win and have the jackpot divided evenly among them in this scenario. So what's the expected rate of return in terms of J and T?

Title: Re: Powerball Expected Value
Post by rmsgrey on Nov 28^th, 2012, 5:41am

Here in the UK, the National Lottery puts half of each ticket's price into the prize fund, meaning, in the long run, you lose half the money you spend on tickets.

In the short run, a £1 ticket is worth slightly less than 50p most of the time because sometimes no-one wins the jackpot - the un-won jackpot rolls over to the next week, making roll-over draws worth more.

Title: Re: Powerball Expected Value
Post by marsh8472 on Nov 28^th, 2012, 9:41am

Here's what I come up with:

[hide](COMBIN(5,5) x COMBIN(34,1) x 1,000,000 + COMBIN(5,4) x COMBIN(54,1) x COMBIN(1,1) x 10,000 + COMBIN(5,4) x COMBIN(54,1) x COMBIN(34,1) x 100 + COMBIN(5,3) x COMBIN(54,2) x COMBIN(1,1) x 100 + COMBIN(5,3) x COMBIN(54,2) x COMBIN(34,1) x 7 + COMBIN(5,2) x COMBIN(54,3) x COMBIN(1,1) x 7 + COMBIN(5,1) x COMBIN(54,4) x COMBIN(1,1) x 4 + COMBIN(5,0) x COMBIN(54,5) x COMBIN(1,1) x 4 - 169,721,370 x 2) / ( COMBIN(59,5) x COMBIN(35,1) ) + summation Y = 1 to T of COMBIN(T, Y) x ((1/175,223,510)^Y x ((175,223,510 - 1)/175,223,510))^(T-Y) x J / Y) [/hide]

Title: Re: Powerball Expected Value
Post by whize on May 29^th, 2013, 4:54pm

By the dogs... what an expression! What language is that? I mean, can I copy paste this to see the result on a computer...?

Title: Re: Powerball Expected Value
Post by towr on May 29^th, 2013, 10:36pm

combin(a,b) is a way to write a!/b!/(a-b)! It's the number of combinations to pick b somethings from a somethings (without replacement).
For example combin(4,2) = 6 because you have only the following 6 combination with 2 numbers picked from 1..4: (1,2) (1,3) (1,4) (2,3) (2,4) and (3,4)
I don't know whether "combin" is a standard way to put it, I see "choose" or just "C" more often. It might be from a specific mathematical programming language.