We’ve heard it: people that invest on the stock market or that gamble in lotteries, casinos, etc usually say “I’m going through a bad patch” (or bad spell). That is, they have been losing money for a while, but hey! better times are ahead and there’s no reason to quit. Are they sure? Are better times ahead? How close is “ahead” to today? Let’s work through a specific example to see how far is “ahead”. Suppose we play a fair game: we toss a coin and with probability 1/2 we get $1 (heads) and with probability 1/2 we lose $1 (tails). We play the game \(n\) times and compute our capital \(C(n)\) up to time \(n\). If our initial capital is zero, then we expect that our capital fluctuate around zero as the coin-tossing game goes on. Sometimes we will be in the “winning area”, where our capital is positive \(C(n)\) > 0. However, we can also be in the “losing area” in which our capital is negative \(C(n)\) < 0. If we are going through a bad patch (being in the losing area) we expect that waiting long enough we will recover and come back to the winning area.
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