Lotteries, State

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State lotteries give participants the dream of winning huge jackpots while providing the state with funding for education, transportation, and other projects. State lotteries currently operate in about three-fourths of the states in the United States. These lotteries provide individuals with a relatively inexpensive means of gambling on the possibility of winning a huge jackpot. The first legal state lottery was established in New Hampshire in 1964, culminating 10 years of legislative effort.

A wide range of games is offered by the numerous state lotteries across the country. The offerings of the Florida State Lottery in 2001, for example, include Florida Lotto, Mega Money, Fantasy 5, Play 4, and CASH 3, as well as numerous instant games played with scratch-off tickets. These games cater to the preferences of potential players by varying the size of the prizes awarded and the probabilities of winning.

Florida Lotto, the premier game in the Florida State Lottery, holds drawings twice a week. In this game, six balls are drawn from a container holding fifty-three balls numbered from 1 to 53. To win the jackpot, the six numbers chosen by a player must match the numbers on the six balls drawn. The order in which the numbers are selected does not matter. The Florida State Lottery claims that the odds of winning this jackpot are 1 in 22,957,480. Players can also win by matching five, four, or three numbers as well. The probability of matching five of six is 1 in 81,410, and continues to increase as the amount of matched numbers decreases.

The probability of winning the jackpot in this game can be verified by determining the amount of six-number combinations possible or by multiplying the probabilities of six consecutive successful selections. Calculating the number of combinations yields the following, which can be expressed using the notation of factorials, denoted as "!",

Of these combinations, only one corresponds to selecting all six numbers correctly. Alternatively, using probabilities, you could multiply the probabilities that the numbers on a ticket are selected on each of the six successive selections. The probability of the first number drawn matching one of the player's chosen numbers is 6 in 53. If this occurs, there are five favorable numbers in the remaining fifty-two, and so on. Therefore, the probability of six consecutive successful selections is or 1 in 22,957,480.

Verifying the probability of correctly selecting five of the six numbers is only slightly more complex. Using the same logic as in the previous example, the probability of the first five matching numbers being selected on the first five draws is . If this occurs, there are forty-eight numbers remaining, and forty-seven are favorable to the result of correctly matching five of six numbers. Thus the probability of attaining this outcome would be . Though the fractions being multiplied would be different, the probability of the first number not matching and the next five matching would be which is equal to the previous result. It is easily verifiable that the probability of the nonmatching number occurring on any of the six selections is equal. Thus the overall probability of correctly selecting five of six is or 1 in 81,409.50355, which rounds to 1 in 81,410.

Fantasy 5 requires the player to match five numbers selected from 26 numbers. Prizes are also awarded to those who correctly match three or four numbers. The probability calculations for this game are analogous to those of Florida Lotto. In Fantasy 5, the probabilities of matching five, four, or three numbers are 1 in 65,780, 1 in 627, and 1 in 32, respectively.

At the other end of the spectrum is Florida's CASH 3 game. This game gives the player a probability of 1 in 1,000 of winning $500 on a $1 ticket or $250 on a 50-cent ticket. In order to win, the player must match a 3-digit number, each digit of which is a number from 0 to 9. In this game, unlike Lotto, each digit is randomly selected from its own set of numbers from 0 to 9 and the order of the result is important.

The Mega Money game adds a bit of a twist. In this game the player picks four numbers from the thirty-two numbers on the top of the ticket and one number from thirty-two on the lower half of the ticket. In order to win, all four numbers on the upper portion and the number on the lower portion of the ticket must be drawn. Finding the probability of matching the first four numbers can be found in the same way as determining the jackpot probability in Lotto. In Mega Money, the probability is in 35,960. To determine the probability of winning the big prize in this game, you must multiply that result by the chances of correctly matching the one number on the lower part of the ticket, which is drawn from another bin of thirty-two balls. The probability of winning the big prize, which averages $200,000 in this game, is: in 1,150,720.

When there are no winners in a large game, like Florida Lotto, new tickets are sold for the next drawing, increasing the value of the prize. Some feel that if the jackpot has not hit several times in a row, that it is "due to be hit." In fact, the probability of any particular combination winning does not change. However, the increase in interest and, correspondingly, in the number of tickets purchased makes it more likely that the jackpot will hit than if a smaller number of tickets were sold.

Consider a hypothetical game in which the probability of winning the jackpot is 1 in 1,000,000. If the jackpot is relatively low and only 100,000 tickets are sold, the probability of the jackpot being hit is at most 1 in 10. The probability is not necessarily equal to since almost certainly, more than one person will have selected the same combination. Now assume the jackpot has not hit for several drawings and a huge prize has accrued. If more than a million tickets have been sold covering 900,000 of the possible 1,000,000 combinations, the probability of the jackpot being hit will now be 90 percent. The chances of any individual ticket winning, however, remains 1 in 1,000,000. An interesting consequence here is that there is no guarantee that a winner will receive the entire jackpot. It is possible that two or more ticket holders will have chosen the correct combination of numbers and would then divide the jackpot equally.

The potential revenue generated by a lottery has provided a strong incentive for states to pass legislation to legalize them. While the exact figures vary from state to state, normally about thirty to thirty-eight percent of the intake goes toward funding state programs. A little more than half is returned in prize money and the remainder is applied to various expenses associated with operating the lottery, such as advertising and paying commissions to vendors. To put the monetary amounts from lotteries that go into state budgets into perspective, the New Hampshire state lottery provided the state department of education with over $65 million in one fiscal year, bringing the total amount of aid to education in that state to $665 million. California boasts of more than $12 billion being earmarked for its public schools since 1985, while the New York state lottery provided $1.35 billion to education in the 1999–2000 fiscal year.

The allure of huge jackpots influences many individuals to purchase lottery tickets. Advertisements touting multi-million dollar jackpots are common. Information regarding the payoff procedures for jackpots is readily available from state lottery commissions as well as from other sources. Most lotteries allow the winner to choose between one immediate lump sum payment or yearly payments over a period of time.

A jackpot winner in the California Super Lotto Plus, for example, can opt for an immediate payment of roughly half of the jackpot amount or can take payment annually for 26 years. In the long-term plan, the first installment is 2.5 percent of the jackpot amount. Successive yearly payments increase each year, with the final payment being about twice the initial. The sum of these twenty-six payments will be equal to the originally stated jackpot amount. Thus the winner of a $6 million jackpot could take approximately $3 million immediately or could get $150,000 as a first installment with successive yearly payments increasing each year and totaling $6 million after all the payments have been made.

Of course, lottery winners must take into consideration the taxes that must be paid on their winnings. Calculating taxes at one-third, a $6 million jackpot winner choosing the one-time payment option would get approximately $4 million. An individual may find that tax savings may be realized in the long term payoff; however, the large lump sum would not be available for investing purposes.

Probability, Experimental; Probability, Theoretical.

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