5-1 Probability Distributions 213 Rationale for Formulas 5-1 Through 5-4 Instead of blindly accepting and using formulas, it is much better to have some understanding of why they work. When computing the mean from a frequency distribution, f represents class frequency and N represents population size. In the expression that follows, we rewrite the formula for the mean of a frequency table so that it applies to a population. In the fraction f>N, the value of f is the frequency with which the value x occurs and N is the population size, so f>N is the probability for the value of x. When we replace f>N with P1x2, we make the transition from relative frequency based on a limited number of observations to probability based on infinitely many trials. This result shows why Formula 5-1 is as given earlier in this section. m = Σ1f # x2 N = ac f # x N d = ac x # f Nd = Σ3x # P1x24 Similar reasoning enables us to take the variance formula from Chapter 3 and apply it to a random variable for a probability distribution; the result is Formula 5-2. Formula 5-3 is a shortcut version that will always produce the same result as Formula 5-2. Although Formula 5-3 is usually easier to work with, Formula 5-2 is easier to understand directly. Based on Formula 5-2, we can express the standard deviation as s = 2Σ31x - m2 2 # P1x24 or as the equivalent form given in Formula 5-4. TABLE 5-5 Roulette Event x P(x) x~ P1x2 Lose -$5 37>38 -$4.868421 Win (net gain) $175 1>38 $4.605263 Total -$0.26 (rounded) (or -26.) b. Craps Game The probabilities and payoffs for betting $5 on the pass line in craps are summarized in Table 5-6. Table 5-6 also shows that the expected value is Σ3x # P1x24 = -7.. That is, for every $5 bet on the pass line, you can expect to lose an average of 7.. TABLE 5-6 Craps Game Event x P(x) x~ P1x2 Lose -$5 251>495 -$2.535353 Win (net gain) $5 244>495 $2.464646 Total -$0.07 (rounded) (or -7.) The $5 bet in roulette results in an expected value of -26. and the $5 bet in craps results in an expected value of -7.. Because you are better off losing 7. instead of losing 26., the craps game is better in the long run, even though the roulette game provides an opportunity for a larger payoff when playing the game once. INTERPRETATION YOUR TURN. Do Exercise 29 “Expected Value for the Florida Pick 3 Lottery.”
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