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Derive the equation for T(x), the mean time that you are in the game when dX is given by Eq. (7.20). Solve this equation for the boundary conditions T(0) = T(C) = 0.

When m is very small, we expect that the solution of Eq. (7.25) should be close to Eq. (7.17) because then the biased game is almost like a fair one. Show that this is indeed the case by Taylor expansion of the exponentials in Eq. (7.25) for m ! 0 and show that you obtain our previous result. If you have more energy after this, do the same for the solutions of T(x) from Exercises 7.5 and 7.2.

Before moving on, let us do one additional piece of analysis. In general, we expect the casino limit C to be very large, so that 2mC ^ 1. Dividing numerator and denominator of Eq. (7.25) by e2mC gives

with the last approximation coming by assuming that eā€”2mC ^ 1. Now let us take the logarithm to the base 10 of this approximation to u(x), so that log10(w(x)) = ā€” 2m(C ā€” x)log10e. I have plotted this function in Figure 7.7, forx = 10 and C = 50, 500, or 1000. Now, C = 1000, x = 10, and m = 0.01 probably under-represents the relationship of the bank of a casino to most of us, but note that, even in this case, the chance of reaching the casino limit before going broke when m = 0.01 is about 1 in

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