In the United States, total annual lottery sales have grown into the billions of dollars.
Lotteries also provide an excellent opportunity to use elementary financial mathematics, as well as some probability, in a context familiar to students. Most students do not have major financial decisions to make, so the principles of financial mathematics may seem far removed from their lives. However, most of them are familiar with the lottery and the topic readily engages them. The application of these mathematical concepts in the lottery is discussed in this paper.
In several USA states and Canada Provinces, the 6/49 lottery operates as an average lottery. To win the lottery grand prize the contestant needs to select all six numbers exactly as drawn in the weekly or monthly contest. This will be used as the model system for the computations in this paper.
Starting with a bag of 49 differently-numbered lottery balls, there is clearly a 1 in 49 chance of predicting the number of the first ball selected from the bag. Accordingly, there are 49 different ways of choosing that first number. When the draw comes to the second number, there are now only 48 balls left in the bag (because the balls already drawn are not returned to the bag), so there is now a 1 in 48 chance of predicting this number.
Thus, each of the 49 ways of choosing the first number has 48 different ways of choosing the second. ...
This continues until the sixth number has been drawn, giving the final calculation, 49 48 47 46 45 44, which can also be written as :
The order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, there are 6 5 4 3 2 1 = 6! or 720 ways they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816. This can also be written as:
This function is called the combination function, denoted as COMBIN(n, k) in some spreadsheet. Taken as a class, the number of possible combinations for a given lottery can be referred to as the "number space" (n). "Coverage" is the percentage of a lottery's number space that is in play for a given drawing (k).
Mathematical concepts can also be applied in making strategies in picking a number. Frequency analysis is a popular strategy that is used. It involves keeping track of the individual numbers that are drawn over a period of time. You might compare it to handicapping a racehorse; rating his past performance to determine what his chances of winning are in the future. The figure below shows how many times each of the numbers has come up in the main National Lottery draw. Here we look at whether the observed distribution of the number of times each of the 49 numbers has come up fits with what would be expected with a truly random draw.
Figure 1. Number of occurrences of 1 to 49 in 1240 lottery draws (UK National Lottery, 1997).
Numbers that appear often in a certain game are called hot numbers. Some players will play these hot numbers exclusively on the assumption that since they have appeared often in the past, they should appear again in the future.