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Wahoo
09-24-2004, 05:23 PM
I have run searches for this topic but have not found any that show the math in a way I can use. I have tried the calculations myself based on what I have found without success. Given the following facts:

35-person tournament with $55 going to prize pool for each.
Payout of 4x(first): 3x(second): 2x(third): 1x(fourth)
Hero is 1.5 times better than average.

I believe the EV for each tournament played will be $82.5
I believe the variance and the standard deviation are very high (71464 and 267 respectively), but this is where my math breaks down, so I have no confidence in these numbers. How can Hero use these numbers (or the correct numbers) to analyze play over time in this tournament?

I believe Hero should expect to be in the money 17% of the time. How do I calculate the percentage chance that Hero will be out of the money after x events. For example, what formula is used to calculate the percentage chance that Hero will play in 8 tournaments and not come in the money?

Thanks for any help,
Wahoo
BruceZ
09-24-2004, 11:10 PM
[ QUOTE ]
35-person tournament with $55 going to prize pool for each.
Payout of 4x(first): 3x(second): 2x(third): 1x(fourth)
Hero is 1.5 times better than average.

I believe the EV for each tournament played will be $82.5

[/ QUOTE ]

You forgot to subtract your losses, and you counted your own buy-in as part of the winnings. Assuming that we place in each of the 4 money positions equally often, our average net winnings when we place in the money are 55*35/4 - 55 = 426.25. An average opponent places in the money with probability 4/35, and since we place 1.5 times more often than average, we place in the money with probability 6/35.

EV = 6/35 * 426.25 + 29/35 *(-55) = $27.50.


[ QUOTE ]
I believe the variance and the standard deviation are very high (71464 and 267 respectively), but this is where my math breaks down, so I have no confidence in these numbers.

[/ QUOTE ]

I get variance = 40,229.93 and standard deviation = 200.57.

Total pot = 55*35 = $1925

1st place net win: 4*192.5 - 55
2nd place net win: 3*192.5 - 55
3rd place net win: 2*192.5 - 55
4th place net win: 1*192.5 - 55
out of money: -55

variance = E(x^2) - E(x)^2

E(x^2) is the expected value of the square of your win or loss. To compute this, square the net win for each place, multiply the squared wins by the probability of each place, and sum. The probability of each place in the money is 6/35 * 1/4. The probability of being out of the money is 29/35.

E(x^2) =
(6/35)*(1/4)*[ (4*192.5 - 55)^2 +
(3*192.50 - 55)^2 +
(2*192.50 - 55)^2 +
(1*192.50 - 55)^2 ] +
(29/35)*(-55)^2

= 41,593.75

E(x) is your EV = 36.93, so E(x)^2 = (36.93)^2

variance = 41,593.75 - (36.93)^2 = 40,229.93.

standard deviation = sqrt(46,279.93) = $200.58.


[ QUOTE ]
How can Hero use these numbers (or the correct numbers) to analyze play over time in this tournament?

[/ QUOTE ]

After N tournaments, you will win an average of N*$36.93 with a standard deviation of sqrt(N)*$200.58. This gives you a range of winnings for various confidence intervals. See my post in this archived thread for more information about standard deviation (http://archiveserver.twoplustwo.com/showflat.php?Cat=&Board=genpok&Number=344985).


[ QUOTE ]
I believe Hero should expect to be in the money 17% of the time. How do I calculate the percentage chance that Hero will be out of the money after x events.

[/ QUOTE ]

(1-0.17)^x


[ QUOTE ]
For example, what formula is used to calculate the percentage chance that Hero will play in 8 tournaments and not come in the money?

[/ QUOTE ]

(0.83)^8 = 22.5%.
BruceZ
09-25-2004, 01:35 AM
[ QUOTE ]
I get variance = <font color="red">40,229.93</font> and standard deviation = <font color="red">200.57</font>.

[/ QUOTE ]

I get variance = <font color="blue">40,837.50</font> and standard deviation = <font color="blue">202.08</font>.



[ QUOTE ]
E(x) is your EV = <font color="red">36.93</font>, so E(x)^2 = (<font color="red">36.93</font>)^2

variance = 41,593.75 - (<font color="red">36.93</font>)^2 = <font color="red">40,229.93</font>.

standard deviation = sqrt(<font color="red">46,279.93</font>) = <font color="red">$200.58</font>.

[/ QUOTE ]

E(x) is your EV = <font color="blue">27.50 as we computed in the first section</font>, so E(x)^2 = (<font color="blue">27.50</font>)^2

variance = 41,593.75 - (<font color="blue">27.50</font>)^2 = <font color="blue">40,837.50</font>.

standard deviation = sqrt(<font color="blue">40,837.50</font>) = <font color="blue">$202.08</font>


[ QUOTE ]
After N tournaments, you will win an average of N*<font color="red">$36.93</font> with a standard deviation of sqrt(N)*<font color="red">$200.58</font>.

[/ QUOTE ]

After N tournaments, you will win an average of N*<font color="blue">$27.50</font> with a standard deviation of sqrt(N)*<font color="blue">$202.08</font>.

Sorry about that. Somehow I copied the wrong standard deviation, and then used that to compute the variance.

Also, note that your EV would be correct if this were a freeroll tourney (no buy-in). That wouldn't change the standard deviation though. It would still be $202.08.
BruceZ
09-25-2004, 01:43 AM
[ QUOTE ]
35-person tournament with $55 going to prize pool for each.
Payout of 4x(first): 3x(second): 2x(third): 1x(fourth)
Hero is 1.5 times better than average.

I believe the EV for each tournament played will be $82.5

[/ QUOTE ]

You forgot to subtract your losses, and you counted your own buy-in as part of the winnings. Assuming that we place in each of the 4 money positions equally often, our average net winnings when we place in the money are 55*35/4 - 55 = 426.25. An average opponent places in the money with probability 4/35, and since we place 1.5 times more often than average, we place in the money with probability 6/35.

EV = 6/35 * 426.25 + 29/35 *(-55) = $27.50.

Your EV would be correct if this were a freeroll tourney (no buy-in).


[ QUOTE ]
I believe the variance and the standard deviation are very high (71464 and 267 respectively), but this is where my math breaks down, so I have no confidence in these numbers.

[/ QUOTE ]

I get variance = 40,837.50 and standard deviation = 202.08.

Total pot = 55*35 = $1925

1st place net win: 4*192.5 - 55
2nd place net win: 3*192.5 - 55
3rd place net win: 2*192.5 - 55
4th place net win: 1*192.5 - 55
out of money: -55

variance = E(x^2) - E(x)^2

E(x^2) is the expected value of the square of your win or loss. To compute this, square the net win for each place, multiply the squared wins by the probability of each place, and sum. The probability of each place in the money is 6/35 * 1/4. The probability of being out of the money is 29/35.

E(x^2) =
(6/35)*(1/4)*[ (4*192.5 - 55)^2 +
(3*192.50 - 55)^2 +
(2*192.50 - 55)^2 +
(1*192.50 - 55)^2 ] +
(29/35)*(-55)^2

= 41,593.75

E(x) is your EV = 27.50, so E(x)^2 = (27.50)^2

variance = 41,593.75 - (27.50)^2 = 40,837.50.

standard deviation = sqrt(40,837.50) = $202.08.


[ QUOTE ]
How can Hero use these numbers (or the correct numbers) to analyze play over time in this tournament?

[/ QUOTE ]

After N tournaments, you will win an average of N*$27.50 with a standard deviation of sqrt(N)*$202.08. This gives you a range of winnings for various confidence intervals. See my post in this archived thread for more information about standard deviation (http://archiveserver.twoplustwo.com/showflat.php?Cat=&amp;Board=genpok&amp;Number=344985).


[ QUOTE ]
I believe Hero should expect to be in the money 17% of the time. How do I calculate the percentage chance that Hero will be out of the money after x events.

[/ QUOTE ]

(1-0.17)^x


[ QUOTE ]
For example, what formula is used to calculate the percentage chance that Hero will play in 8 tournaments and not come in the money?

[/ QUOTE ]

(0.83)^8 = 22.5%.
Wahoo
09-25-2004, 03:18 PM
Thanks for your time on this. It is exactly what I needed. I can work the math with the formulas, but was lost trying to put it together myself.

Thanks,
Wahoo