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I like to play Blackjack and I am currently learning a new card counting system. I am studying the according betting strategy right now and there are graphs for certain Risks of Ruin. However, to determine the values I want I would have to estimate them from the graph because there are no equations given for the functions of the graphs.

Can anybody help me determine those functions for the graphs?

This are the graphs:

enter image description here

And here is a table for the $~5\%~$ - Curve:

\begin{array} {|r|r|} \hline x & f(x) \\ \hline 0.5 & 240 \\ \hline 1 & 150 \\ \hline 1.5 & 110 \\ \hline 2 & 75 \\ \hline 2.5 & 65 \\ \hline 3 & 55 \\ \hline 3.5 & 45 \\ \hline 4 & 38 \\ \hline 4.5 & 35 \\ \hline 5 & 32 \\ \hline \end{array}

I already thought of $~f(x) = k/x~$ where $~k =~ 150~$, but it's very unprecise so I wanted to have a quite exact function.

nmasanta
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Simon Eggersdorfer
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    Do you realise that there are infinitely many functions that pass through these points? It would be wise to limit the search space to certain functions only. You mentioned $$ f(x) =\frac{k}{x} $$ so this is a good starting point. But perhaps we need a correction term? – Matti P. Mar 23 '20 at 12:52
  • Another way is to do a Langrange or spline interpolation. – EuklidAlexandria Mar 23 '20 at 12:58
  • @MattiP.

    I understand what you mean. You can see on the picture how the graphs look like and the function is only important for 0 < x <= 10

     – Simon Eggersdorfer Mar 23 '20 at 13:40
  • @EuklidAlexandria

    I am not that good at math to understand what that means ^^

     – Simon Eggersdorfer Mar 23 '20 at 13:40
  • My opinion is that $f(x)=150/x$ is not such a bad approximation. But to get an even closer result, it would be very beneficial to know some theory. Is there any theory behind how the values were obtained? – Matti P. Mar 23 '20 at 13:42
  • Never mind, in this case the solution proposed by Matti P. is better. Basically, Lagrange interpolation is a polynomial going through all points, whereas spline interpolation is a piece-wise polynomial going through all points. – EuklidAlexandria Mar 23 '20 at 13:44
  • Well, I can send you the pdf (the book) which I am reading and where the graphs and other things can be found. – Simon Eggersdorfer Mar 23 '20 at 13:47

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