3 Probability
Where We’ve Been
Identified the objective of inferential statistics: to make inferences about a population on the basis of information in a sample
Introduced graphical and numerical descriptive measures for both quantitative and qualitative data
Where We’re Going
Statistics in Action Lotto Buster! Can You Improve Your Chance of Winning?
“Welcome to the Wonderful World of Lottery Busters.” So began the premier issue of Lottery Buster, a monthly publication for players of the state lottery games. Lottery Buster provides interesting facts and figures on the 42 state lotteries and 2 multistate lotteries currently operating in the United States and, more importantly, tips on how to increase a player’s odds of winning the lottery.
In 1963, New Hampshire became the first state in modern times to authorize a state lottery as an alternative to increasing taxes. (Prior to that time, beginning in 1895, lotteries were banned in America because they were thought to be corrupt.) Since then, lotteries have become immensely popular, for two reasons. First, they lure you with the opportunity to win millions of dollars with a $1 investment, and second, when you lose, at least you believe that your money is going to a good cause. Many state lotteries, like Florida’s, earmark a high percentage of lottery revenues to fund state education.
The popularity of the state lottery has brought with it an avalanche of “experts” and “mathematical wizards” (such as the editors of Lottery Buster) who provide advice on how to win the lottery—for a fee, of course! Many offer guaranteed “systems” of winning through computer software products with catchy names such as Lotto Wizard, Lottorobics, Win4D, and Lotto-Luck.
For example, most knowledgeable lottery players would agree that the “golden rule” or “first rule” in winning lotteries is game selection. State lotteries generally offer three types of games: Instant (scratch-off tickets or online) games, Daily Numbers (Pick-3 or Pick-4), and the weekly Pick-6 Lotto game.
One version of the Instant game involves scratching off the thin opaque covering on a ticket with the edge of a coin to determine whether you have won or lost. The cost of the ticket ranges from 50¢ to $1, and the amount won ranges from $1 to $100,000 in most states to as much as $1 million in others. Lottery Buster advises against playing the Instant game because it is “a pure chance play, and you can win only by dumb luck. No skill can be applied to this game.”
The Daily Numbers game permits you to choose either a three-digit (Pick-3) or four-digit (Pick-4) number at a cost of $1 per ticket. Each night, the winning number is drawn. If your number matches the winning number, you win a large sum of money, usually $100,000. You do have some control over the Daily Numbers game (since you pick the numbers that you play); consequently, there are strategies available to increase your chances of winning. However, the Daily Numbers game, like the Instant game, is not available for out-of-state play.
To play Pick-6 Lotto, you select six numbers of your choice from a field of numbers ranging from 1 to N, where N depends on which state’s game you are playing. For example, Florida’s current Lotto game involves picking six numbers ranging from 1 to 53. The cost of a ticket is $1, and the payoff, if your six numbers match the winning numbers drawn, is $7 million or more, depending on the number of tickets purchased. (To date, Florida has had the largest state weekly payoff, over $200 million.) In addition to capturing the grand prize, you can win second-, third-, and fourth-prize payoffs by matching five, four, and three of the six numbers drawn, respectively. And you don’t have to be a resident of the state to play the state’s Lotto game.
In this chapter, several Statistics in Action Revisited examples demonstrate how to use the basic concepts of probability to compute the odds of winning a state lottery game and to assess the validity of the strategies suggested by lottery “experts.”
Statistics in Action Revisited
Recall that one branch of statistics is concerned with decisions made about a population on the basis of information learned about a sample. You can see how this is accomplished more easily if you understand the relationship between population and sample—a relationship that becomes clearer if we reverse the statistical procedure of making inferences from sample to population. In this chapter, we assume that the population is known and calculate the chances of obtaining various samples from the population. Thus, we show that probability is the reverse of statistics: In probability, we use information about the population to infer the probable nature of the sample.
Probability plays an important role in making inferences. Suppose, for example, you have an opportunity to invest in an oil exploration company. Past records show that, out of 10 previous oil drillings (a sample of the company’s experiences), all 10 came up dry. What do you conclude? Do you think the chances are better than 50:50 that the company will hit a gusher? Should you invest in this company? Chances are, your answer to these questions will be an emphatic “No!” However, if the company’s exploratory prowess is sufficient to hit a producing well 50% of the time, a record of 10 dry wells out of 10 drilled is an event that is just too improbable.
Or suppose you’re playing poker with what your opponents assure you is a well-shuffled deck of cards. In three consecutive five-card hands, the person on your right is dealt four aces. On the basis of this sample of three deals, do you think the cards are being adequately shuffled? Again, your answer is likely to be “No,” because dealing three hands of four aces is just too improbable if the cards were properly shuffled.
Note that the decisions concerning the potential success of the oil-drilling company and the adequacy of card shuffling both involve knowing the chance—or probability—of a certain sample result. Both situations were contrived so that you could easily conclude that the probabilities of the sample results were small. Unfortunately, the probabilities of many observed sample results aren’t so easy to evaluate intuitively. In these cases, we need the assistance of a theory of probability.