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Online poker tournament statistics : probability of winning a poker tournament

Tuesday, April 7th, 2009 | Mark | Poker Articles, Poker Mathematics, Poker Tournament

This article is a part of the Poker Mathematics series.

Recently I have wondered if it is possible to calculate the probability of winning a poker tournament based on which strategy you use and how your all in moves are distributed. In this article I will share with you my findings. Hope you have the patience to read it all the way through:-). The table below (the culmination of way too many hours of research) is a teaser of what is ahead if you continue reading.

Duration [hours]	Reference win rate [1 out of ..]	On a roll win rate [1 out of..]	Coinflip win rate [1 out of ..]	Underdog win rate [1 out of..]	Realistic win rate [1 out of..]
1	9	2	8	37	na
2	60	3	32	412	16
3	140	4	64	1372	28
4	340	5	128	4572	50
5	680	6	256	15242	87
6	1110	7	512	50805	152

Check out the previous post in this series:

• Online poker tournament statistics: player exit percentage
• Online poker tournament statistics: cummulative blinds

Just to remind you, in my first post in this series I discovered an exponential relationship between the time an online freezeout tournament has been running and the percentage of players exited from the tournament. This enabled me to estimate both the time needed to reach the final table of a tournament given the number of players registered and what size tournament you should choose given the time you have available to play.

I needed this relationship to be able to estimate the number of players entering a tournament of a given duration and thus the total amount of chips in play. Once I know the total amount of chips in play is known, I can calculate the number of successive all in wins needed to win all the chips in the tournament. Finally, the last step in my calculations will  be to set up different all in probability distributions to end up with the probability of winning an online poker tournament. Confused? Don’t worry:-) I will guide you through my calculations step by step in the remainder of this article. If you don’t like math, simply scroll down to the final table where I summarize my most important findings.

The table below summarizes the total amount of chips in play for different tournament durations and the successive all ins you will need to win to win the tournament:

Tournament duration [hours]	Number of Players	Total Chips in play (starting stack 1500)	Successive all ins needed to win (rounded numbers)
1	9	13500	3
2	60	90000	6
3	140	210000	7
4	340	510000	8
5	680	1020000	9
6	1110	1665000	10

Again, in order to keep things simple I assumed that each all in would double the  1500 chip starting stack. According to this assumption, 2 successive all in wins will increase your stack from 1500 to 6000, 3 successive all inn wins from 1500 to 12000 and so on.

To make things a bit more realistic, let’s assume that you win half the chips you need to win the tournament by making your opponent fold (i.e no show down). In this case the number of  successive all ins needed to win is reduced by 1 for each of the tournament durations shown above.

I have chosen the following scenarios that IMO cover the typical all in situations you will experience during a tournament.

The simple reference calculation

Setting probability calculations aside and assuming all the players in the tournament (including yourself) have an equal chance of winning it, you will win a tournament with x registered players 1 out of x times. This means you will win a 100 player tournament 1 out of 100 times, a 200 player tournament 1 out of 200 times and so on. Obviously you should aim higher than this otherwise your bankroll will hit zero in no time.

1st scenario: On a roll

• You have an 80% probability of winning all your hands.
• If for example you need to win 3 successive all ins to win the tournament, the probability of winning it is 0,8*0,8*0,8 = 0,5 corresponding to 1 out of 2 tournaments.

2nd scenario: Coinflip

• You have a 50% probability of winning all your hands.

3rd scenario: Underdog

• You have a 30% probability of winning all your hands.

4th scenario: Realistic?

• 20% of your hands you are underdog (30% probability)
• 40% of your hands you have a coinflip (50% probability)
• 40% of your hands you are favorite (80% probability)

The table below summarizes the probability calculations for the 4 scenarios:

Duration [hours]	Reference win rate [1 out of ..]	On a roll win rate [1 out of..]	Coinflip win rate [1 out of ..]	Underdog win rate [1 out of..]	Realistic win rate [1 out of..]
1	9	2	8	37	na
2	60	3	32	412	16
3	140	4	64	1372	28
4	340	5	128	4572	50
5	680	6	256	15242	87
6	1110	7	512	50805	152

Interestingly, it turns out that in all but one scenario (the underdog) your tournament win rate is significantly larger than the simple reference calculation. In conclusion I think I have successfully managed to give a qualified estimate of the probabilities of winning online poker tournament. In my next article I will try to make some use of the numbers I have come up with. An obvious approach would be to calculate the expected return on investment (ROI) for each of the scenarios listed above.

I would greatly appreciate any comments on the math and final numbers.

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Tags: Poker Articles, Poker Mathematics, Poker Tournament

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