By Marek Fisz

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**Example text**

The potential payoff associated with the “×” option is mind-numbingly large, albeit highly unlikely. 7 × 10113), and we tend to appreciate this in terms of its intrinsic worth, or utility, rather than its actual value. So, we compare the expected utilities of the two options, rather than the expected payoffs. To simplify the discussion we use the common (base 10) logarithm of a payoff as a measure of its utility. (The common logarithm of a positive real number is the power to which 10 must be raised to yield the number.

Up or down? If, say, 1,000 such tacks are tossed and 160 land point up, we could invoke Bernoulli’s law of large numbers and assume the true probability is approximately equal to the experimental probability of 160/1000 = 4/25 or 16 percent. A more reliable estimate could be achieved by tossing a larger number of tacks and recording the results. In practice empirical probabilities are used as estimates of true probabilities because the true (theoretical) probabilities are not generally accessible.

Doing the Right Thing 43 is the case for all casino games; otherwise, the casino would lose money and go out of business. For Keno the house advantage can be as large as 27 percent, and for slot machines it varies between 4 percent and 8 percent.

### Probability Theory and Mathematical Statistics, Third Edition by Marek Fisz

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