The process we want to model then is that we call $20 before the flop: 88 percent of the time we miss the flop and fold, 10 percent of the time we go to the river and win, 2 percent of the time we go to the river and lose.
If we go to the river, we will put an extra $90 in the pot. That assumes that there is a raise and reraise on the flop, one raise on the turn, and one bet on the river.
To see how implied odds affect the outcome, you can modify assumptions I’m making about pot size and analyze this with different assumptions about the action that will occur after the flop.
With these assumptions the range of outcomes is that we can lose $20 (88 percent of the time), lose $110 (2 percent of the time), or win $190 (10 percent of the time). The expected value is then
0.88 x (-20) + 0.02 X (-110) + 0.10 X 190 = 0.6]
This means that calling a raise with only four other callers will lost about $60 on the average.
This result is sensitive to assumptions about how these players play after the flop and how large the pots will get after the flop. We can calculate the variance of this situation as
(0.6) ² -0.88 x (-20)(-0.20) + 0.02 x (-110) (-110) + 0.10 x (190) (190)
or (352 + 242 + 3610) – 0.36 = 4203.64
Notice that the variance is a very large value. We often use the square root of the variance, which gives us a value in scale with the expected value.
The square root of the variance called the “standard deviation. ”In this example the standard deviation is 64.9.
A large variance is common in poker situations. In this case it’s interesting to note that most of the variance comes from our 10 percent of large wins.
This demonstrates that having a large variance is not always a bad thing. To use variance to reflect risk, we can use what’s called a semi-variance.
That’s computed by adding up only the terms in the variance that result from negative outcomes. In this example the semi-variance is 352 + 242 = 594, with a semi-standard deviation of 24.
Risk is a fundamental element of gambling. Without risk, there is no gamble. Whenever you have uncertain outcomes, you have risk.
The usual view of risk is that the more uncertainty about the outcome, the more risk.
Sometimes you’ll see a perspective that the more likely the outcome is to result in a loss, the higher the risk.
This view of risk would suggest that buying a lottery ticket is high risk because the probability that your ticket will lose is very high.
This is the kind of result that you get if you use variance as a proxy for risk, but as we’ve just seen, it’s usually a better idea to use semi-variance as a measure of risk.
STATISTICS
Statistics is about the analysis of actual results-data analysis. It’s kind of science of information reduction.
Statistical analysis involves estimating a small group of parameters that somehow gives a description of the distributional properties of a large set of data.
The usual parameters of interest are the mean and variance.
You can compute these from actual rather than from the probability distributions.
This book is not intended as a primer on data analysis, and you should refer to any standard text on statistical methods for the formulas to use in analyzing actual data.
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