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Exact calculation: Push

The first two examples are all about gaining a proper understanding of the mathematics behind the independent chip model. In the first example, we will evaluate a push step by step in the same way that programs like SNG Power Tools do.

Example:

55$ SNG, 4-handed, Blinds 300/600

CO: 6000

BU: 4000 (Hero)

SB: 4000

BB: 6000

CO folds. Hero has 22. Push or Fold?


1. Estimation of the Opposing Calling-Range

First, we estimate our opponent's calling range:

SB: 88+, A8+

BB: 88+, A8+


2. How much are the chips worth?

Now we calculate how much our chips are worth in actual dollars. To do this, we must determine the with what probability we place in each of the top three places:



It's easy for first place:


P(1st place) = Hero's Chips / Total Chips = 4000 / 20000 = 0.2 = 20%

So Hero Has a 20% chance of placing first, not taking position and skill into account.

It is a bit more difficult to make this same calculation for second and third place. We must assume in turn that one of the other three players has taken first place and then calculate the probability that hero wins against the remaining players. It goes like this:


P(2nd place) = P(CO takes 1st) * Heros Chips / (Total Chips - Number Chips CO) + P(SB takes 1st) * Heros Chips / (Total Chips - Number Chips SB) + P(BB takes 1st) * Heros Chips / (Total Chips - Number Chips BB) = 0.3 * 4000 / (20000 - 6000) + 0.2 * 4000 / (20000 - 4000) + 0.3 * 4000 / (20000 - 6000) = 0.0857 + 0.05 + 0.0857 = 0.2214 = 22.14%

The calculation for third place is messy but works analogously.


P(3rd place) = 0.2642 = 26.42%

The probability of our placements multiplied by the payoff for that place and summed over the first three places gives us the true value of our chips:

EV(T4000) = P(1st place) * $(1st place) + P(2nd place) * $(2nd place) + P(3rd place) * $(3rd place) = 0.2 * $250 + 0.2214 * $150 + 0.2642 * $100 = $109.63 or 21.9% ($109/$500)