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Lotteries are among the simplest and most widely played of all games of chance, and unfortunately for the gambler, among the worst in terms of expected value. Lotteries come in such an incredible number of variations that it is impractical to analyze all of them. Thus, in this section, we will study some of the more common lottery formats.
The basic lottery is a random experiment in which the gambling house (in many cases a government agency) selects numbers at random, without replacement, from the integers from 1 to . The integer parameters and vary from one lottery to another, and of course, cannot be larger than . The order in which the numbers are chosen usually does not matter, and thus in this case, the sample space of the experiment consists of all subsets (combinations) of size chosen from the population .
Recall, or show that
.Naturally, we assume that all such combinations are equally likely, and thus, the chosen combination , the basic random variable of the experiment, is uniformly distributed on .
The player of the lottery pays a fee and gets to select numbers, without replacement, from the integers from 1 to . Again, order does not matter, so the player essentially chooses a combination of size from the population . In many cases , so that the player gets to choose the same number of numbers as the house. In general then, there are three parameters in the basic lottery.
The player's goal, of course, is to maximize the number of matches (often called catches by gamblers) between her combination and the random combination chosen by the house. Essentially, the player is trying to guess the outcome of the random experiment before it is run. Thus, let denote the number of catches.
Show that the number of catches in the , lottery has probability density function
The distribution of is the hypergeometric distribution with parameters , , and , and is studied in detail in the chapter on Finite Sampling Models. In particular, from this section, it follows that the mean and variance of the number of catches is
Note that if or However, in most lotteries, and is much larger than . In these common cases, the density function is positive for the values of given in Exercise 2.
We will refer to the special case where as the , lottery; this is the case in most state lotteries. In this case, the probability density function of the number of catches is
The mean and variance of the number of catches in this special case are
Explicitly give the probability density function, mean, and standard deviation of the number of catches in the lottery.
Explicitly give the probability density function, mean, and standard deviation of the number of catches in the lottery.
Explicitly give the probability density function, mean, and standard deviation of the number of catches in the lottery.
The analysis above was based on the assumption that the player's combination is selected deterministically. Would it matter if the player chose the combination in a random way? Thus, suppose that the player's selected combination is a random variable taking values in . (For example, in many lotteries, players can buy tickets with combinations randomly selected by a computer; this is typically known as Quick Pick). Clearly, and must be independent, since the player (and her randomizing device) can have no knowledge of the winning combination . In the following exercise, you will show that such randomization makes no difference.
Let denote the number of catches in the lottery when the player's combination is a random variable, independent of the winning combination . Show that has the same distribution as in Exercise 1. Hint: condition on the value of .
There are many web sites that publish data on the frequency of occurrence of numbers in various state lotteries. Some gamblers evidently feel that some numbers are luckier than others.
Given the assumptions and analysis above, do you believe that some numbers are luckier than others? Does it make any mathematical sense to study historical data for a lottery?
The prize money in most state lotteries depends on the sales of the lottery tickets. Typically, about 50% of the sales money is returned as prize money; the rest goes for administrative costs and profit for the state. The total prize money is divided among the winning tickets, and the prize for a given ticket depends on the number of catches . For all of these reasons, it is impossible to give a simple mathematical analysis of the expected value of playing a given state lottery. Note however, that since the state keeps a fixed percentage of the sales, there is essentially no risk for the state.
From a pure gambling point of view, state lotteries are bad games. In most casino games, by comparison, 90% or more of the money that comes in is returned to the players as prize money. Of course, state lotteries should be viewed as a form of voluntary taxation, not simply as games. The profits from lotteries are typically used for education, health care, and other essential services. A discussion of the value and costs of lotteries from a political and social point of view (as opposed to a mathematical one) is beyond the scope of this project.
Many state lotteries now augment the basic , format with a bonus number. The bonus number is selected from a specified set of integers, in addition to the combination , selected as before. The player likewise picks a bonus number , in addition to a combination . The player's prize then depends on the number of catches between and , as before, and in addition on whether the player's bonus number matches the random bonus number chosen by the house. We will let denote the indicator variable of this latter event. Thus, our interest now is in the joint distribution of .
In one common format, the bonus number is selected at random from the set of integers , independently of the combination of size chosen from . Usually . Note that with this format, the game is essentially two independent lotteries, one in the , format and the other in the , format.
Explicitly compute the joint probability density function of for the lottery with independent bonus number from 1 to 27. This format is used in the California lottery, among others.
Explicitly compute the joint probability density function of for the lottery with independent bonus number from 1 to 42. This format is used in the Powerball lottery, among others.
In another format, the bonus number is chosen from 1 to , and is distinct from the numbers in the combination . To model this game, we assume that is uniformly distributed on , and given , is uniformly distributed on the set of combinations of size chosen from . For this format, the joint probability density function is harder to compute.
Show that
Condition on whether or not is in to show that
Explicitly compute the joint probability density function of for the lottery with bonus number chosen as described above. This format is used in the Super 7 Canada lottery, among others.
Keno is a lottery game played in casinos. For a fixed (usually 80) and (usually 20), the player can play a range of basic games, as described in the first subsection. Typically, ranges from 1 to 15, and the payoff depends on and the number of catches . In this section, you will compute the density function, mean, and standard deviation of the random payoff, based on a unit bet, for a typical keno game with , , . The payoff tables are based on the keno game at the Tropicana casino in Atlantic City, New Jersey.
Recall that the probability density function of the number of catches , from Exercise 2 above, is given by
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 |
---|---|---|
Payoff | 0 | 3 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 |
---|---|---|---|
Payoff | 0 | 0 | 12 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 |
---|---|---|---|---|
Payoff | 0 | 0 | 1 | 43 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|
Payoff | 0 | 0 | 1 | 3 | 130 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 1 | 10 | 800 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 1 | 4 | 95 | 1500 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 1 | 25 | 350 | 8000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 0 | 9 | 90 | 1500 | 25,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 0 | 4 | 50 | 280 | 4000 | 50,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 0 | 1 | 22 | 150 | 1000 | 5000 | 100,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 80 | 400 | 2500 | 25,000 | 100,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 32 | 200 | 1000 | 5000 | 25,000 | 100,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 20 | 80 | 600 | 3500 | 10,000 | 50,000 | 100,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 9 | 42 | 310 | 1100 | 8000 | 25,000 | 50,000 | 100,000 |
The payoff table for is given below. Compute the probability density function, mean, and standard deviation of the payoff.
Catches | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Payoff | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 25 | 100 | 300 | 2800 | 25,000 | 50,000 | 100,000 | 100,000 |
In the exercises above, you should have noticed that the expected payoff on a unit bet varies from about 0.71 to 0.75, so the expected profit (for the gambler) varies from about to This is quite bad for a casino game, but as always, the lure of a very high payoff on a small bet for an extremely rare event overrides the expected value analysis for most players.
With , show that the top 4 prizes (25,000, 50,000, 100,000, 100,000) contribute only about 0.017 (less than 2 cents) to the total expected value of about 0.714.
On the other hand, the standard deviation of the payoff varies quite a bit, from about 1 to about 55.
Although the game is highly unfavorable for each , with expected value that is nearly constant, which do you think is better for the gambler--a format with high standard deviation or one with low standard deviation?