Lotteries, in various legal and illegal incarnations, have always been around. During times of prosperity people play them for fun and during difficult financial times, such as the one the planet is going through right now, people play them with the hopes of improving their current condition. It is beyond the scope of this article to discuss the ethical implications of these government run institutions. Instead, I will attempt to show you as concisely and simply as possible how you, yourself, can have a little fun calculating the odds of hitting it big. This is a neat skill that you can apply to many other things in everyday life so I believe this is a useful exercise.
In order to get to the more complicated lotteries we have to ease into the topic by examining the simpler types. Such simple lotteries are the popular three and four drum set lotteries usually known by the trademarked names Cash 3 and Play 4 or other variant of these. In these types of games we have separate drums containing separate sets of balls numbering from 0 through 9. The object of these games is to match the number drawn in the right order. Therefore, ORDER DOES MATTER! In order to calculate the odds of hitting the jackpot we note that each ball, in each of the sets, has an equal chance of being drawn. Consequently, for each drum there are 10 distinct possibilities and since the order matters, we have to multiply together all the possible outcomes per drum in order to obtain the total number of possible outcomes. Therefore, for the three drum three ball set games we have:
Since there is only one way to match all the numbers in the right order we find that the odds of winning is 1 in 1000 also written as 1:1000. Similarly for the four ball set games we have:
In other words, there are 10,000 distinct outcomes in these games and the player has a 1 in 10,000 (1:10,000) chance of hitting it big.
These types of simple games, which place emphasis on the order of numbers, belong to a subset of combinatorics called permutations. Combinatorics by the way is the branch of mathematics that deals with the behavior of finite and countable sets of objects.
More complicated games, on the other hand, belong to the subset of combinations. In combinations the order of the numbers DOES NOT MATTER. That is to say, if you have the numbers 1 and 2 to choose from they can be ordered as the two distinct permutations {1,2} and {2,1}. (Or another way to think about it is that if you have two pool balls labeled 1 and 2, you can pick up ball 1, then ball 2 or pick up ball 2 then ball 1.) However, since the order doesn’t matter for combinations those two are counted as one combination. Similarly, the number set {1,2,3} can be ordered as the distinct permutations:
{1,2,3}
{1,3,2}
{2,1,3}
{2,3,1}
{3,1,2}
{3,2,1}
That means that there are six ways to arrange three objects. (Note, that I use the word object because you have to keep in mind for later that these three objects can stand for distinct numbers, or for distinct pool balls, or even different dogs.) However, as before there is only one combination that includes all of these permutations. If we were to continue down the line expanding sets, following the same pattern as before, we would soon realize that the number of permutations per single combination of distinct objects is equal to:
n!,
where n stands for the number of objects in the set and ! is the factorial symbol. To take the factorial of number mean to multiply all consecutive numbers from the given number all the way to one. For example:
3! = 3 x 2 x 1 = 6
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040
1! = 1
0! = 1, this a special case and it is agreed to be equal to one.
Another important step is to realize the reason why this is true. This can be illustrated with a quick thought exercise. If we had five different pool balls and we picked them up one by one, each time there would be one less total number of possible outcomes. So we have:
5 pool balls, pick 1 = 5 choices to pick from
4 pool balls, pick 1 = 4 choices to pick from
.
.
1 pool ball, pick 1 = 1 choice to pick from
When multiply the number of choices (or distinct outcomes) available each time, in order to find the total number of possibilities, we arrive back at our nifty n!
Finally armed with all this knowledge we are able to tackle the more complicated pick-6 (like most state lotteries, Powerball, and MegaMillions) or pick-7 (like the EuroMillions) games. In this example we will use the Florida Lottery and the Powerball to aid us in our illustration of the calculations. We will choose the Florida Lottery first because it is the less complicated of the two and because once we learn the principles behind the calculations, we can easily apply them to the Powerball and in fact any other finite set game.
Well, let’s being! At the time of writing the Florida lottery consists of a single set of 53 white balls labeled 1 through 53. In order to win the jackpot one must match the 6 balls drawn. If we apply what we learned before, we would easily realize that the way to calculate the total probable number of combinations is to multiply the probability of picking each of the six balls, at the time they are picked, and to divide this number by the total number of ways a set of six (winning) balls can be arranged. Therefore, we have:
When you examine the number above you might, “wait a minute… this number does not match the 22,957,480 number of total combinations forecasted on the lottery website!” Of course you would be right. A lottery with 1 in nearly 15.5 billion would be impossible to win. What you have failed to realize is that that number in the number of permutations! Remember that as we said we have to divide that number by the number of ways a set with six objects can be organized. In this case there are six objects, so they can be organized in 6! ways.
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Therefore, the total number of combinations that can be drawn is:
15,529,385,600/720 = 22,957,480
Since there is only one way to win, then we can see that the odds of winning are the 1:22,957,480 predicted on the lottery site!
Before we can proceed to calculate the odds for the lesser prizes or those for the Powerball we must develop the notation for a neat trick known in mathematics as the binomial distribution which will greatly simplify everything to follow. Observe that the binomial distribution can be calculated as follows:
where n is the total number of balls in the set and q is the number of balls we pick. You might recognize the q! term as that term that we must divide out to eliminate all the permutations of each possible combination of balls. But what about the other term? In order to illustrate what the terms n! and (n – q)! stand for we will expand them out using the Florida lottery as an example. i.e. n = 53 and q = 6.
We must notice that all the terms in the multiplication after 47 cancel out, so we are left with,
53 x 52 x 51 x 50 x 49 x 48 = 15,529,385,600
and we can clearly recognize from this that n!/(n – q)! is nothing more but a short-hand generalized way of writing the total number permutations for 6 balls picked from a set of 53! With this knowledge in hand we can finally tackle the probability of winning the Powerball! The Powerball is currently a pick 5 out of 59 and 1 out of 35 lottery. In order to calculate the total number of combinations we calculate the total number of combinations for the 5 out of 59 pick, like this,
and then total number combinations for picking 1 out of 35,
and multiply them together,
5,006,386 x 35 = 175,223,510.
Consequently, since 175,223,510 is the total number of total combinations possible in the lottery and there is only one way to match a single winning combination, the odds of winning the jackpot in Powerball is 1:175,223,510. This method can be applied to calculate the odds of winning the jackpot in any and all lotteries and similar situations involving sets of objects that may arise in everyday life.
Finally, the last important lesson we have left to learn is how to calculate the lower tier prizes. In order to calculate these we have to make the important realization that after a drawing the balls essentially subdivide into two groups… the balls we must match to lose and the balls we must match to win. By multiplying the probabilities of these two groups we obtain the total number of combinations possible. It can be written as follows,
For example, assume that we want to know the probability and odds of matching 5 out of 6 balls in the current setup of the Florida lottery. How would we call calculate this? Well, following the formulation above it would be something like this:
or,
Since the 282 is the total number of probable outcomes, then we have that the odds of buying a ticket with 5 matching numbers out of 6 winning number are 282 in 22,957,480 or 1:81,409.50. (Note: Divide Both sides by the number on the left side of the odds in order to simplify it.)
Similarly, for the Powerball each number set splits into sets of winners and losers and we must multiply all of them together in order to get the total.
Let’s give it try… let us calculate the odds of winning the second highest price in Powerball, for which we must match all the white balls and not match the red powerball. Therefore in this case we have:
The reason why the answer here is 34 should be rather obvious… while there is only one way to match all the white balls, there are 34 ways to not match the red ball. Moreover, the odds of winning the second largest Powerball prize are 34 in 175,223,510 or 1:5,153,632.65.
Well, there you have it! Using what you learned you can now calculate the odds of winning the jackpot and the lower tier prizes for any lottery. Hope you have fun playing with numbers, instead of the lottery itself, so you can put your hard earned money towards something more important.
I suggest you go ahead and calculate the odds for every lottery that you know and then teach your friends how to do by sending them to this article.
Happy crunching!