How do you calculate this?

[SpaceAce]

Let's say you offer someone X:1 that the next hand they are dealt will be pocket Jacks, where X < what you would need to offer to make the proposition 0 EV. How do you calculate how many trials you would need to have a Y% chance of making a profit?

Thanks,
SpaceAce

[Paul2432]

This can be figured using the BINOMDIST function is MS Excel. Set the fields as follows:

Trials = number of trials

Number_S = number of wins your opponent would need to break even based on the odds offered. For example, if you offer 9:1 this should be 10% of the number of trials. In the spreadsheet this should be setup as calculation based on the number of trials.

Probability_s = probability of a success based on the true odds.

Cumulative = True

The result of the function is the value you are looking for. You can either manually adjust the number of trials, or use goal seek so that the number of trials matches your desired percentage.

There is no easy way to do this by hand.

Paul

[SpaceAce]

OK, thanks for the reply.

[ QUOTE ]

There is no easy way to do this by hand.


[/ QUOTE ]

Bummer [img]/images/graemlins/frown.gif[/img] I wasn't as interested in getting the result as in learning the proper procedure to follow for solving the problem.

Thanks,
SpaceAce

[AaronBrown]

I think Paul2432 gave you the correct procedure. If the trial and error part worries you, just start with one trial and keep adding one until you get to the correct result. If using Excel bothers you, there is a formula to compute this by hand.

You can get a good approximate result without the trial and error. You win X 1 time in 221 and lose 1 220 times in 221. That makes your expected value per bet (X - 220)/221. Your standard deviation per bet is (X + 1)*SQRT[220]/221.

The ratio of expected value over standard deviation is (X - 220)/[(X + 1)*SQRT(220)]. The table below shows you the value you need for various levels of Y. For example, for a 60% chance of making a profit, you need expected value over standard deviation to be 0.25 or more. You can get these numbers from a Normal probability table or using the Excel NORMSINV function.

60% 0.25
80% 0.84
90% 1.28
95% 1.64
99% 2.33
99.9% 3.09
99.99% 3.72

To figure out the number of trials needed, take the critical value from the table above, divide it by the per bet value you computed, and square the result. For example, if X is $225, your per-bet value is 5/[226*14.83] = 0.001492.

To get 99% safety, you need [3.09/0.001492]^2 = 4,292,218 bets.