3.2. RUIN PROBABILITIES 101
This says that if the working capital is much greater than the amount required
for victory, then the probability of ruin is reasonably small.
Then the probability of an unbroken string of ten successes in ten years
is:
(1− 1 /10)^10 ≈exp(−1)≈ 0. 37This much success is reasonable, but simple psychology would suggest the
gambler would boast about his skill instead of crediting it to luck. More-
over, simple psychology suggests the gambler would also blame one failure
on oversight, momentary distraction, or even cheating by the casino!
Another Interpretation as a Random Walk
Another common interpretation of this probability game is to imagine it as a
random walk. That is, we imagine an individual on a number line, starting
at some positionT 0. The person takes a step to the right toT 0 + 1 with
probabilitypand takes a step to the left toT 0 −1 with probabilityq and
continues this random process. Then instead of the total fortune at any
time, we consider the geometric position on the line at any time. Instead of
reaching financial ruin or attaining a financial goal, we talk instead about
reaching or passing a certain position. For example, Corollary 3 says that if
p≤q, then the probability of visiting the origin before going to infinity is 1.
The two interpretations are equivalent and either can be used depending on
which is more useful. The problems below are phrased in the random walk
interpretation, because they are more naturally posed in terms of reaching
or passing certain points on the number line.
The interpretation as Markov Processes and Martingales
The fortune in the coin-tossing game is the first and simplest encounter with
two of the most important ideas in modern probability theory.
We can interpret the fortune in our gambler’s coin-tossing game as a
Markov process. That is, at successive times the process is in various
states. In our case, the states are the values of the fortune. The probability
of passing from one state at the current timetto another state at timet+ 1
is completely determined by the present state. That is, for our coin-tossing