212 CHAPTER 5 Discrete Probability Distributions Not Exactly, but “At Least as Extreme” It should be obvious that among 1000 tosses of a coin, 502 heads is not significantly high, whereas 900 heads is significantly high. What makes 900 heads significantly high while 502 heads is not significantly high? It is not the exact probabilities of 900 heads and 502 heads (they are both less than 0.026). It is the fact that the probability of 502 or more heads 10.4622 is not low, but the probability of 900 or more heads 10+2 is very low. It is unlikely that we would get 252 or more heads in 460 coin tosses by chance. It follows that 252 wins by teams that won the overtime coin toss is significantly high, so winning the coin toss is an advantage. This is justification for changing the overtime rules, as was done in 2012. INTERPRETATION YOUR TURN. Do Exercise 19 “Using Probabilities for Significant Events.” PART 2 Expected Value and Rationale for Formulas Expected Value In Part 1 of this section we noted that the expected value of a random variable x is equal to the mean m. We can therefore find the expected value by computing Σ3x # P1x24, just as we do for finding the value of m. We also noted that the concept of expected value is used in decision theory. In Example 6 we illustrate this use of expected value with a situation in which we must choose between two different bets. Example 6 involves a real and practical decision. Be a Better Bettor EXAMPLE 6 You have $5 to place on a bet in the Golden Nugget casino in Las Vegas. You have narrowed your choice to one of two bets: Roulette: Bet on the number 7 in roulette. Craps: Bet on the “pass line” in the dice game of craps. a. If you bet $5 on the number 7 in roulette, the probability of losing $5 is 37>38 and the probability of making a net gain of $175 is 1>38. (The prize is $180, including your $5 bet, so the net gain is $175.) Find your expected value if you bet $5 on the number 7 in roulette. b. If you bet $5 on the pass line in the dice game of craps, the probability of losing $5 is 251>495 and the probability of making a net gain of $5 is 244>495. (If you bet $5 on the pass line and win, you are given $10 that includes your bet, so the net gain is $5.) Find your expected value if you bet $5 on the pass line. Which of the preceding two bets is better in the sense of producing higher expected value? a. Roulette The probabilities and payoffs for betting $5 on the number 7 in roulette are summarized in Table 5-5. Table 5-5 also shows that the expected value is Σ3x # P1x24 = -26.. That is, for every $5 bet on the number 7, you can expect to lose an average of 26.. SOLUTION
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