152 CHAPTER 4 Probability Identifying Significant Results with Probabilities: The Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of a particular observed event is very small and the observed event occurs significantly less than or significantly greater than what we typically expect with that assumption, we conclude that the assumption is probably not correct. We can use probabilities to identify values that are significantly low or significantly high as follows. Using Probabilities to Determine When Results Are Significantly High or Significantly Low ■ Significantly high number of successes: x successes among n trials is a significantly high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. That is, x is a significantly high number of successes if P1x or more2 … 0.05.* ■ Significantly low number of successes: x successes among n trials is a significantly low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less. That is, x is a significantly low number of successes if P1x or fewer2 … 0.05.* *The value 0.05 is not absolutely rigid. Other values, such as 0.01, could be used to distinguish between results that are not significant and those that are significantly high or low. See Example 1 on pages 144–145, which illustrates the following: ■ Among 20 births, 20 girls is significantly high because the probability of 20 or more girls is 0.000000954, which is less than or equal to 0.05 (so the gender selection method appears to be effective). ■ Among 20 births, 12 girls is not significantly high because the probability of 12 or more girls is 0.251722, which is greater than 0.05 (so the gender selection does not appear to be effective). Probability Review ■ The probability of an event is a fraction or decimal number between 0 and 1 inclusive. ■ The probability of an impossible event is 0. ■ The probability of an event that is certain to occur is 1. ■ Notation: P1A2 = the probability of event A. ■ Notation: P1A2 = the probability that event A does not occur. PART 2 Odds Expressions of likelihood are often given as odds, such as 50:1 (or “50 to 1”). Here are advantages of probabilities and odds: ■ Odds make it easier to deal with money transfers associated with gambling. (That is why odds are commonly used in casinos, lotteries, and racetracks.) ■ Probabilities make calculations easier. (That is why probabilities tend to be used by statisticians, mathematicians, scientists, and researchers in all fields.) Id T Gambling to Win In the typical state lottery, the “house” has a 65% to 70% advantage, since only 30% to 35% of the money bet is returned as prizes. The house advantage at racetracks is usually around 15%. In casinos, the house advantage is 5.26% for roulette, 1.4% for craps, and 3% to 22% for slot machines. The house advantage is 5.9% for blackjack, but some professional gamblers can systematically win with a 1% player advantage by using complicated card-counting techniques that require many hours of practice. If a card-counting player were to suddenly change from small bets to large bets, the dealer would recognize the card counting and the player would be ejected. Card counters try to beat this policy by working with a team. When the count is high enough, the player signals an accomplice who enters the game with large bets. A group of MIT students supposedly won millions of dollars by counting cards in blackjack. I s “ 6 a s t
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