Ruin Again: Part 3 – Blackjack

In my August article I explained how to take an arbitrary random variable X and approximate it as a random walk.  The two statistical quantities we used were the mean M and the variance.  Using the material in my September article we could then find an approximate solution to the ruin problem.

Let’s suppose that X represents the random variable depicting the wager in Blackjack when the initial bet is one unit.  I am assuming a six-deck game with doubling on any first two cards, splitting up to three times except for Aces, dealer hits the soft seventeen, and the player uses Basic strategy.  The expected return M for such a game is approximately -0.0058.  The variance V for generic Blackjack was calculated by Peter Griffin in his classic book The Theory of Blackjack.  Peter’s figure was 1.26.

Now let us suppose that the player makes initial wagers of size w.  What are the mean MM and variance VV in this case?  This is easily settled as follows.  Let Y = wX.  Then Y represents the random variable of wagers when the initial bet is w.  Hence from (1) in my August article we have

                                    MM = E(wX) =wE(X) = wM   (1)

Similarly using (3) from my August article we have

                        VV = E(Y²) – MM² = E((wX)²) – (wM)²

or

                        VV = E(w²X²) – w²M² = w²(E(X²) – M²) = w²V   (2)

The bet size b is given by (4) of my August article and is

                        b = sqrt(w²V + w²M²) = wsqrt(V + M²)        (3)

where sqrt is the square root function.  Now for Blackjack the term M² is so tiny compared to V that we can ignore it so that

                        b = wsqrt(V)

The square root of V is called the standard deviation and is denoted by S.  Hence we finally have

                        b = wS   (4)

Finally the winning and losing probabilities p and q for the random walk are given by (5) and (6) of my August article and are

                        p = ½ + wM/2b  and q = ½ - wM/2b

According to my September article we need the ratio p/q.  From the calculation above this ratio is just

                        p/q =(1/2 + wM/2b)/(1/2 – wM/2b)

Factoring ½  out of the top and bottom of this fraction (they cancel) we are left with

                        p/q = (1 +wM/b)/(1 –wM/b)

which can be written as

                        p/q = (b +wM)/(b – wM)

Substituting  wS for b in this last expression and factoring out w from the top and bottom we have

                        p/q = (S + M)/(S – M)    (5)

Let us assume that our initial bankroll is B and we wish to gain an amount G.  Then a in relation (7) of my September article is B + G and a – x is just G.  Thus we have

ruin probability ={ [(S + M)/(S – M)]^G/wS -1}/{[(S + M)/S – M)]^{(G + B)/wS} -1}   (6)

S = standard deviation for a game using unit bets

M = mean of the game using unit bets

B = bankroll

G = attempted gain

w = wager size

An expression similar to (6) was first introduced by Patrick Sileo (The Evolution of Blackjack Games Using a Combined Expectation and Risk Measure) www.bjmath.com/bjmath/sileo/sileo.pdf

For Blackjack S = 1.1225 and M = 0.0058.  If we risk 10 units to win 3 then B = 10, G = 3 and w = 1.  Using these numbers in (6) I was able to calculate the ruin probability to be approximately 0.241.  How good is this approximation?  Well, I programmed a simulation and played 50 million hands.  The ruin my program gave me was 0.257; the approximation was a 6.21% error.  You be the judge.  See you next month

Don Catlin can be reached at 711cat@comcast.net