Wednesday, August 15, 2007

Not All Lottery Combinations are Equal

If you play the lottery, invest in stocks, begin a business, play bridge, poker, backgammon, or simply want to be successful in a random world, you need to understand the odds of an event occurring in your particular sub-world, rather than simply relying on luck. By knowing the probabilities, you can make sound judgments and take advantage of the principals of randomness instead of being subject to them. Successful people utilize probability theory to take a particular viewpoint and then make their financial investments accordingly. This post takes the controversial viewpoint that all lottery combinations are not equal, and examines the probabilities that certain types of combinations will be drawn sooner than others. Winning is not gambling, it is all about mathematics.

For years, my friend Ron and I have argued about the best combinations for winning the Mega Millions and Powerball lotteries. I believe that the most likely numbers to be drawn are those which are spread across the range of numbers. Ron insists that all combos have an equal chance of occurring and always plays the classic: 1 2 3 4 5 and 6. Since neither of us has won the jackpot yet, we continue to have this debate. However, after launching Lottery Power Picks last year, I am more convinced than ever that playing the numbers 1-2-3-4-5-6 is a poor investment and not worth the risk. This article explores why:
  • All combinations do NOT have an equal chance of occurring.
  • The odds of the combination 1,2,3,4,5,6 being drawn is minimal.
Powerball Example
For purposes of this discussion, we will focus on the Powerball. In this lottery, a player selects 5 numbers from a pool of 55 numbered white balls, and one Powerball from a set of 42 red balls. When the drawing is later held on Wednesday or Friday night, there will be two drums, one containing the 55 white balls, and one with the 42 red balls. The five white balls will be drawn first, one at a time, without replacement, which means there will no duplicate white ball numbers. Immediately afterwards, the single Powerball will be drawn. These six numbers become the winning combination for that drawing (Understanding Powerball Chances). Since there are 3,478,761 unique white ball combinations and 42 red balls, there is a grand total of 146,107,962 six number combinations. Thus, the overall chances of any single combination occurring is 1 in 146.1 million.

So then all Combinations are Equal?
On the surface, the answer is yes, because when the final combination is drawn, it will truly be the only 1 of the 146.1 million combinations. Thus, it would appear that any other combination could have been equally selected. This is why the official lotteries, most casual players, and lottery skeptics, like Ron, argue that all combinations are equal.

Not So
Both mathematicians and seasoned, or regular, lottery players know that: based on probabilities, certain combinations are more likely to be drawn than others. This means that all combinations are NOT equal. The argument for this hypothesis is that lottery drawings are completely random. Remember: "random does not mean haphazard. Even though individual outcomes are uncertain, it is possible to predict distributions of random events if we know their probability" (Randomness). We shall use these principals as the basis for our argument.

But First, Forget the Powerball

The Powerball Lottery actually consists of two separate drawing in one. The first is the drawing of the 5 white balls, and the second is the drawing of the Powerball itself. Since there is only one Powerball drawn out of a set of 42, the likelihood of any one ball being picked is identical. Thus, it really doesn't matter what the Powerball is. A player can easily eliminate this Powerball risk by playing all 42 Powerballs with each 5 number combination. Therefore, the Powerball is not important to our discussion, and we shall ignore it.

Concentrate on Predicting the 5 White Ball Combination
The primary objective of the lottery player is to pick 5 numbers before the drawing which will match those numbers chosen in the actual drawing. Lets examine two possible 5 white ball combinations, 01 02 03 04 05 and 01 19 22 38 53 to see if they have an equal chance of winning. For these purposes, we will use the Lottery Power Picks Combination Distribution Tables as our reference guidelines in order to evaluate the likelihood that the combination will be picked.

First, let's review the scenario when the player selects the combination: 1, 2, 3, 4, 5. These numbers do not look random because have a pattern and a sequence. However, these traits are not important in the mathematics of randomness. A number sequence does not have to "look" random in order to be random. Instead, it must follow well established criteria.
  1. First, from an even-odd viewpoint, we see that the sequence 1, 2, 3, 4, 5 contains 2 even numbers and 3 odd numbers. There are 1,149,876 of these combinations and have a 33.1% chance of being selected (Table PB-1a: Powerball Even/Odd Combination Distribution). This is the best even/odd choice. Good so far.
  2. Second, we examine this combination's bucket distribution. Here we notice that all five numbers fall within the Bucket 1-9. From Table PB-2a: Powerball Bucket Distribution, we find that there are only 126 combinations in this bucket, meaning its probability of occurrence is 0.00362% (126/3,478,761). This means that we will have a 1 in 126 chance of winning once every 27,609 drawings or 265.5 years. Not good.
  3. Third, we check the combination's hi-low distribution in Table PB-3a: Powerball Low-Hi Number Distribution. Reading across the table with a first ball of 1 and a last ball of 5, we learn that there is only 1 possible outcome in the set of 3,478,761 combinations. From this viewpoint, the probability of this sequence being drawn is even closer to 0%, only 1/3,478,761. However, in this case, we can expect to be 100% correct once every 3,478,761 drawings or 33,449.6 years.
Comparatively, let's review the scenario where the player selects the numbers 1, 19, 22, 38, 53 instead.
  1. Since this combination also has 2 even numbers and 3 odd numbers, it too has a 33.1% chance of being selected. This is the same as the above sequence.
  2. Looking at this number's bucket distribution, we find that there are 54,000 combinations that can be formed using the numbers in Buckets 1-9, 10-19, 20-29, 30-39, and 50-55. This set of numbers represents 1.55% of all combinations, meaning that we can expect this condition to occur approximately once every 65 drawings or 32.5 weeks. When this happens, our odds of winning will be improved to 1/54000.
  3. Next, after checking the combination's hi-low distribution of 1-53, we discover that there are 20,825 possible combinations having a 0.59% probability of occurring. This means that once every 167.0 drawings or 1.6 years, we will have a 1 in 20825 chance of winning.
Probabilities Favor Certain Combinations
From this information, we have learned how to understand the Lottery Power Picks Probability Distribution tables. From them, we can argue that one combination is more likely to be drawn than another because of its probabilities. Further, we have observed that:
  1. The combination 1 2 3 4 5 has an extremely low probability of occurring.
  2. Other combinations, such as 1 19 22 38 53, have higher probabilities of occurring.
But, before we can say this definitively, we need to examine the past Powerball results to see if it really does follow the probabilities we have asserted.

Verify Probabilities using Historical Results
Using the above two numerical sequences, we have reviewed and calculated the actual distribution probabilities by summarizing the Lottery Power Picks Powerball Results from January 01, 2001 through August 11, 2007. During this period, there were 692 drawings. Of these, we observed that:
  • 227 combinations contained 3 odd and 2 even balls which is 32.8%. Using this set's 33.1% probability, we expected to find 228.7 combinations. (both sequences)
  • 0 combinations were all contained in bucket 0-9 which is 0%. Using this set's 0% probability, we expected to find 0 combinations. (1-2-3-4-5 only)
  • 0 combinations had a high-low distribution of 1-5 which is 0%. Using this set's 0% probability, we expected to find 0 combinations. (1-2-3-4-5 only)
  • 7 combinations had a bucket distribution of 0-9, 10-19, 20-29, 30-39, 50-55 which is 1.01%. Using this set's 1.55% probability, we expected to find 10.7 combinations. (1-19-22- 38-53 only)
  • 4 combinations had a high-low distribution of 1-53 which is 0.58%. Using this set's 0.59% probability, we expected to find 4.1 combinations. (1-19-22- 38-53 only)
As expected, the actual combination frequencies are very similar to the expected frequencies, verifying our assumption that: we can reasonably predict the distribution of Powerball lottery outcomes based on various distribution probabilities.

Conclusion: Not All Lottery Combinations are Equal
Given the fact that the Powerball and other lottery drawings are truly random, we have illustrated that it is possible to predict certain lottery outcomes knowing the likelihood, or probability, of that particular subset. Having applied the mathematical principals of randomness to two different sets of lottery numbers, we have shown that one combination is more likely to be drawn than another.

Since we have demonstrated that two combinations have different likelihoods of occurring, we may confidently conclude that we have proven the statement that:

All lottery combinations do NOT have an equal chance of occurring.

Not All Lottery Combinations are Equal!


Special Note - Not True for Pick 3 and Pick 4
This proof does not apply to Pick 3 and Pick 4 lotteries. In these, all combinations are truly equal for two reasons. First, the final combination is order dependent. Second, the numbers are picked with replacement, meaning that duplicate digits can occur: 221, 999, 777, etc.

Learn More
You can learn more about this topic by reading the articles listed below.

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