It's getting a bit late, so hopefully this comes out coherent and somewhat intelligible. However, I was a bit intrigued by the Catalan number idea, so I looked over it a little bit.
1) Not sure where you obtained the formula's application, but I think you have misinterpreted this particular application. On your blog, you made the assumption that it applies for
all vertices above the abscissa. However, according to an example I'll link, it only applies for vertices that lie on or above the diagonal line that starts at the origin and ends at the final vertical point (corresponding to xf=14 in your example). So, it actually excludes 1/2 of the possible vertices that should be included above the abscissa.
http://mathforum.org/library/drmath/view/63019.html
(search for the city lawyer problem).
The correct formula for the probability of being always being above water (or survival as you called it) -- more technically, likelihood of 2t vertices occurring on the positive side of the abscissa, is p2t,2n =
(2t choose t)*((2n-2t) choose (n-t))*(1/2^2n). ** In the case of your example of final trial being 2n = 14 (where n = 7 in the formula), the resulting probability of all vertices being positive is 20.9%.
Notice it's about double your result (i.e. the bottom area of box diagonal your application excluded. It's not exactly a linear relationship, as the probabilities follow something called an arcsin law.
Another corollary to the arcsin law (regarding your 50/50 assumption) is that a 50% likelihood of being above the abscissa or in the lead has the least likelihood of occuring (i.e. the balance you refer to is flawed). It is increasingly more likely that one side will lead at any fraction above 50%
.
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Regarding intraday bill's reply, notice the correct formula includes one of the comments you stated -- the divisor is comprised of 2^2n
events (or one over the probability of all possible outcomes with 50/50 chance per event), rather than the 7 factorial and other coefficients used in mdama's blog example.
Regarding your interpretation of survival, it's sort of what I mentioned earlier, that there is a great divide between some of the binary theories being postulated on some of these threads and reality. These types of simple binary problems assume fixed betting with only two binary outcomes.
If trading was really like that, then while there might be relative drawdowns; once you cross the horizontal axis you have blown up -- capital depleted. In this sense, the model is correct -- survival implies you have not crossed below the horizontal axis, or else your capital is gone.
Reality often deals more along the lines of non-parametric distributions with a large range of continuous outcomes, and often they have fat tails, which completely nullify many of these types of binary theories. It also deals with compounding , unlike the example application here, as well as fractional betting. All of these pragmatic issues change the dynamics of this simple model quite a bit.
Probability of survival in dama's model (or even any simple binomial model) is not the same under 14 coin tosses being all heads compared to other possible outcomes. The likelihood of all heads is quite different than say, half heads and tails -- meaning the probability of exceeding the horizontal threshold is likewise different. In two coin tosses, two heads are not equally likely to show up as a head and a tail. Basic binomial distribution.
**The formula is shown in the 1st book I referenced earlier.
