What are the odds of hitting lotto? What are the chances that you will be struck by lightning? What is the possibility that your five children will all be boys? Such questions as these belong to the realm of probability in the field of mathematics. Probability theory was developed to help answer these questions, and the early pioneer mathematicians, Pascal, Fermat, and Huygens, helped steer the way toward a coherent theory. Today probability theory, along with its sister discipline of statistics, helps answer questions relating to voting polls, weather patterns, mortality data, and of course, things like lotto.
Odds and Probability
Probability deals with calculating the likelihood that a specific event will occur. For example, in flipping a fair coin, there are two events: heads and tails. Probability expresses the quotient of the event that is desired, the numerator, to the total number of events, the denominator. Thus in the coin example with two events, the probability of either heads or tails is 1/2.
The odds of an event express the ratio of its occurrence to its non-occurrence. Thus in the coin example, the odds of heads to tails are 1:1. That is, there is one way for heads to occur and one way for it not to occur; that is, when the coin shows tails. Let us take another example: suppose you have 3 red balls and 2 blue balls in a bag. What is the probability of drawing at random a red ball? This is easy to answer. The desired result is red ball, and there are three of them. There is a total of five balls in the bag. The desired probability is thus 3/5. Now we can ask, "What are the odds of drawing a red ball?" The odds are 3:2 in favor of drawing red. Notice that in the ratio 3:2, the 2 is derived by subtracting 3 from 5 in the expressed probability ratio.
Pick 3
What makes probability fascinating is its ability to calculate easily the chances of winning certain gambling games. Take the state Pick 3 game. What is the probability of choosing the winning combination in exact order? In order to answer this question, we need to determine two things: the desired choice, which is our winning combination, and the total number of possibilities, also known as the sample space. The hardest part of any probability problem is calculating either or both of these numbers. Let us say we choose the combination 392. Obviously, there is only one favorable outcome for the numerator of our expression, as we have only played one. The question now is how do we find the sample space, or the denominator?
To do this, we have to employ some counting techniques. The branch of mathematics that treats the counting procedures used in both probability and statistics is called combinatorics, and its theory, combinatorial analysis. This theory teaches us how to count sample spaces using different approaches. To count the sample space for our Pick 3 problem, we observe that there are 10 possible outcomes, namely the digits from 0-9. If we think of three slots, each slot holding a different outcome, then the possible outcomes for slot 1 are 10; the same applies to slots 2 and 3. To get the total sample space, we simply multiply the number of possible outcomes for each slot together. This yields 10x10x10 or 1000. The probability of our pick 3 combination 392 coming out then is 1/1000. Notice here that order of the outcome is critical. That is 392 is not the same as 329.
Lotto
Calculating the probability of lotto is a bit different and requires slightly more work. For this example, we will use the New York standard lotto game, which uses balls numbered 1-59. You select 6 numbers in this range, and you win if your combination comes out (lucky stiff if it does!).
The difference here from the Pick 3 game discussed earlier is that the order in which the numbers comes out does not matter. Thus suppose you choose 2, 6, 32, 39, 41, and 4. In order to calculate the probability of this event occurring, we need to calculate the sample space and the preferred event. To do this, we need to introduce something known as combinations, which is part of combinatorial analysis.
To find the sample space, we resort to the slots we used earlier. The lotto balls from 1-59 are placed one by one into each of six slots. The first slot can be occupied by any of 59 balls. Once this ball is consumed the second slot can only be occupied by 58. The third by 57, the fourth by 56, the fifth by 55, and the sixth by 54. Multiplying these numbers together we get the very large number of combinations 32,441,381,280. This represents the total number of possible ways those 6 numbers can be drawn. Now you see why lotto is so hard to win!
Now for the numerator, we need to find out how many ways your six numbers can be arranged, as order does not matter. This is very similar to the way we computed the sample space using the six slots. Only now we do it with six numbers instead of 59. Thus the first slot can be occupied by any of the six numbers: 2, 6, 32, 39, 41, and 4. The next slot by any of five and so on down to 1. If we multiply 6x5x4x3x2x1, we get 720. The probability is thus 720/32,441,381,280, or 1/45,057,454. In terms of odds, out of 45,057,454 possibilities, you have 1 chance. The odds are 1: 45,057,453. Maybe you should spare the dream and save that buck. But then again, who knows?
References
- Hald, Anders. History of Probability and Statistics and Their Applications Before 1750. New Jersey: John Wiley & Sons, 2003.
- Rumsey, Deborah. Probability for Dummies. Indiana: Wiley Publishing, Inc., 2006.
Joe Pagano - Joe is a prolific writer, author, poet, and linguist, who has published over 200 articles and several books on a broad array of ...
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