Sportsbook probability question - bank units?

[Byrn]

Can someone provide me a formula for the following:

If the bank is "booking" bets at 11:10 (ignoring moneyline bets), I am curious to know how many units (x) the bank should have to give (y) % chance of going broke assuming each bet is one unit and has a 50/50 chance of winning or losing. If there is clearer way to state it or better way to look at it, let me know. Also, is there any way to adjust for moneyline bets (assuming you know your average line wager placed)?

Thanks in advance!

[mmbt0ne]

::I don't guarantee any of this::
I guess it depends on the distribution of winning bets. Assuming the bank has $0 on hand before taking bets, it would need to win 10/21 (about 47.6%) of the bets to cover their losses.
Now, take event X to be the event that the bank makes 1000 bets(for simplicity).
X~Binomial(1000,.50) where .50 is the probability that they lose a bet, and 1000 is the number of bets booked for the week.
Instead of going through complicated cumulative distribution function summations, just use the normal approximation of a binomial.

P(X<=x) = (Phi)(x+0.5-np)/(np(1-p))^.5
P(X<=476) = (Phi){476+.5-(1000)(.5)}/{1000*.5(.5)}^.5
P(X<=476) = (Phi)(-1.48) = 0.0681

So about 6.81% of the time they would go broke assuming that they had no money on hand. The (Phi) function isn't something that I can replicate for you because I just look up the values on a table once I get the value. I don't know where you'd find them on the internet either because I get them out of the back of a book, but if you really want them, I can scan it for you.

Anyone see a problem here?

[Byrn]

Thanks for the input. So if I understand correctly, that is the equivalent of saying that there is a 6.81 % chance of a loss after 1000 bets booked?

I knew those college stat classes would come in handy someday! Is there some excel formula I can use to model this?

[mmbt0ne]

You could easily model everything except the Phi() function just by inputing the number of trials, and probability of the initial event. I guess after that you could just look each number up on a chart to find the actual percentage. I'll see if I can find anything more helpful though.

[mmbt0ne]

Ok, try this.
http://www.f2f2s.com/Applications/Betting.xls

It's not perfect, and could include more, but it should give you a basic idea of the percentages involved. Notice that even at 50% chance of a winning bet for the sportsbook, after 5500 or so bets of equal money, it's almost impossible for them to go broke according to the probabilities.

[Byrn]

[ QUOTE ]
Ok, try this.
http://www.f2f2s.com/Applications/Betting.xls

It's not perfect, and could include more, but it should give you a basic idea of the percentages involved. Notice that even at 50% chance of a winning bet for the sportsbook, after 5500 or so bets of equal money, it's almost impossible for them to go broke according to the probabilities.

[/ QUOTE ]

That spreadsheet is fantastic. Thanks!