The Arithmetic of Lotto-The Fundamental Principle of Counting-TipsRate It [ 0.00 / 0 Votes ]
Any thoughts? "But I already know how to count," you say. I agree; you probably do. And you probably already know this fundamental principle, if not in words at least in concept. But I'll state it here anyway, since it is critical for the kind of counting we have to do when enumerating the number of possible Lotto tickets.
If an event can occur in any of m different ways, and if after that another event can occur in any of n different ways, then the number of ways both events can occur in that order is m times n.
For instance, suppose you are in a room with 3 doors that lead to the interior of a building having 4 exits. Then the number of routes you could take to leave the building is 3 times 4, or 12. As another example, suppose you will randomly select from a committee consisting of 10 people one person to be the chairperson and one to be the vice-chairperson. You have 10 choices for chairperson, but after having made a choice only 9 people remain who can be chosen as vice-chairperson, so there are 10 times 9, or 90 ways to make that two-person selection. If you want to choose a chairperson, a vice-chairperson, and a treasurer, say, you can just extend the fundamental principle to obtain the product (10)(9)(8) = 720 ways to make the three-person selection.
So is that all there is to counting up the number of possible Lotto tickets? For a 40-number "Pick 5" Lotto game, do we just say there are 40 ways to draw the first ball, then 39 for the next, and so on, so that the total number of possible 5-number tickets is just the product (40)(39)(38)(37)(36)? No. According to the fundamental principle, that's the number of ways to draw the balls in that order. But the order in which the balls are drawn is not material in Lotto, only the particular set of numbers that we see, regardless of order. We shall see in the next section, however, how the fundamental principle of counting will allow us to adjust for sets and not orderings.
PLUS Lotto, a Web-based lottery out of Liechtenstein, does offer a variation on Lotto that will yield to our fundamental principle directly. In that game a person must choose two numbers from a set of 40 numbers, with repetition of numbers allowed. Then the PLUS Lotto computer will randomly select two numbers. If the player's two numbers match the two numbers drawn, in order, by the computer, then the player wins the jackpot. For instance, if the player chooses 13 first and 16 second while the computer also chooses 13 first and 16 second, the player wins. But if the computer chooses 16 first and 13 second (or any other ordering, like 24 and 22 or 8 and 8), the player does not win. So for that variation on Lotto, how many possible two-number choices are there? Answer: 40 times 40, or 1600.
And what, my friend, is the probability that the computer will draw at random, in the same order, the same two numbers you select? Correct! Answer: 1 divided by 1600, or we say you have a 1 in 1600 chance of winning the jackpot with one such ticket. If the ticket were to cost you $2, at least how much should the jackpot be before you will be making a bet with favorable odds? If you said $3,200 you already have a good appreciation of gambling.
By Cybergeezer
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<> (Submitted on: Fri Mar 02, 2001) 100
"But I already know how to count," you say. I agree; you probably do. And you probably already know this fundamental principle, if not in words at least in concept. But I'll state it here anyway, since it is critical for the kind of counting we have to do when enumerating the number of possible Lotto tickets.