 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.

Introduction 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.

Therefore:
Not All Lottery Combinations are Equal!

Done.

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.

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1. OF (692) DRAWINGS YOU PUBLISHED (238) OCCURANCES. (4) HIGH/LOW DISTRIBUTIONS. (227) 3 ODD & 2 EVEN COMBINATIONS. AN (7) BUCKET DISTRIBUTIONS . WHAT HAPPENED IN THE OTHER 454 DRAWINGS?! PLEASE RESPOND! THANKYOU VERY MUCH!

2. The other 454 have lottery fever - they're standing in line waiting to buy their Lottery Power Picks for Friday's \$325 Mega Million drawing (excuse my lottery humor)

Seriously, however, you can't add these numbers together because each represents a different statistic - meaning each is a different view of the same data.

It might be helpful if you brought up the Powerball Results Table in a separate window. If you count all the drawings form Jan 1 2001 through Aug 11 2007, you would get 692 drawings.

Next, count all the "2e / 3o" entries in the Even/Odd column for this period. There are 227 of these out of the 692. The others are different.

To find the 4 high/low distributions, find all the results that have #1 as ball one, and #53 as ball 5. All others start and/or end with different numbers.

The 7 bucket distributions can be located by counting the " 111101 " entries in the Buckets column.

As you can see, we are simply summarizing the same set of drawing results differently.

If you added these numbers together, you would count some results more than once. For example, the results on 12/16/06 and 07/02/05 are both in the 2e/3o category and the 111101 buckets.

I'm sorry this wasn't stated more clearly.

Thank-you for reading our article. And, I appreciate you taking the time to question this.

JL...

3. I am with your friend Ron on this one. Let's talk about it a little bit. Admittedly, I am not a mathematician (I am a physicist, though), so bear with me.

I will first try to explain this in less formal terms for your average readers out there. Your general argument seems to be that, because a group of similar combinations (e.g. any combination with 2 even and 3 odd numbers) can occur with relatively higher probability than a different group (e.g. any combination with all odd numbers), then a PARTICULAR combination from the first group (e.g. 1,2,3,4,5) is more likely to appear than a PARTICULAR one from the second (e.g. 7,13,25,45,29).

Let's get more formal: it is true that you can establish equivalence classes, each of which defining a subset of the entire sample space and each of which having different relative probabilities, but that doesn't change that any two combinations out of the original sample space then will occur with EQUAL probability.

In regards to the combination 1,2,3,4,5, you say that:

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.

This math works out and I did verify it. Does this mean that I have a 33.1% chance of winning if I choose 1,2,3,4,5? Not at all. I would have a 33.1% chance of winning if my lottery ticket said on it "you win if a combination with 2 even numbers and 3 odd numbers is drawn". But it doesn't. My lottery ticket says I win if the combination 1,2,3,4,5 is drawn and nothing else. The reason your probability looks large here is that there are many possible combinations with 2 even and 3 odd numbers. A few examples are (1,2,3,4,5), (22,44,13,5,35), (11,27,12,55,54).

You then continue:

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.

You might be confusing people now. You just said that the they had a 33.1% chance of winning with the combination 1,2,3,4,5, but now they have a 0.00362% chance? That is a huge discrepancy isn't it? Your math checks out ok again, but we are talking about a different probability here. This is the probability of getting any combination with all of the balls having numbers between 1 and 9 inclusive. The probability is so much smaller because there are far fewer possible combinations here than the last example. A few possible combinations in this set would be (1,2,3,4,5), (1,4,7,8,9), (2,6,3,7,5). Once again, my ticket says I win if and only if the numbers 1,2,3,4,5 show up, NOT any combination with numbers between 1 and 9 inclusive.

And there is more:

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.

Coincidentally (or not), this equivalence class includes only one possible combination: 1,2,3,4,5. And this probability is the real probability of me winning if my ticket says 1,2,3,4,5.

You then provide another example combination 1,19,22,38,53 against which to compare:

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.

This is the same as the last sequence. The same thing I said last time applies here too.

Continuing:

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.

Yes, of course the probability will be higher when you are choosing from 5 different buckets. This is because lottery draws numbers without replacement. The probability would have been better if you had used the 40-49 bucket in place of the 50-55 because this bucket is bigger. Once again, my lotto ticket would never say "you win if your 5 numbers fall within the buckets 1-9, 10-19, 20-29, 30-39, and 50-55". Instead, it says I win if and only if the numbers 1,19,22,38,53 are drawn.

And another comment on this point. What if you reduced the size of your buckets and centered them on these numbers? What if you had buckets 1-1, 19-19, 22-22, 38-38, and 53-53---i.e. five buckets each with only one number inside of them, centered on the numbers in the combination that you are considering right now? Well, the probability of that occurring is only 1/3,478,761. That is close to 0% just like the last combination. Does this mean that 1,2,3,4,5 and 1,19,22,38,53 are equally likely to show up in a draw? I think so.

Then, you say:

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.

This is just insanity here. I think you should explain more about what this 1-53 hi-low distribution really means. My interpretation of it must be different because this time the math does not check out. As far as I can tell, you are talking about all combinations with the five numbers being between 1 and 53 inclusive. That means that (1,19,22,38,53) falls into this set but (1,19,22,38,54) does NOT due to the number 54. Can you clarify this? Either way, I have a good feeling that everything I have said earlier still applies here.

Finally, you conclude:

The combination 1 2 3 4 5 has an extremely low probability of occurring.

Yes, it does.

Other combinations, such as 1 19 22 38 53, have higher probabilities of occurring.

I don't buy this one. 1 19 22 38 53 occurs with the exact same probability as 1 2 3 4 5: 1/3,478,761 or about 0.000029%. All you have really shown me is that you can partition the sample space up into classes that have different relative probabilities. You chose classes that included these two combinations such that 1 19 22 38 53 looked more favorable, but this does not change the hard fact that the two are equally likely.

One last thing... your "verified results" section have simply verified the probabilities of those equivalence classes that I keep talking about. What you should be verifying 1,2,3,4,5 vs. 1 19 22 38 53. How many times has 1,2,3,4,5 come up and how many times has 1 19 22 38 53 come up since January 2001? I bet both have come up zero times. The mathematics will back my assertion.

4. Jordan,

You have made some very good points, and misunderstood the hi-low distribution. But, let me add some clarification for you.

1. Your introductory summary is correct, my argument is about grouping similar combinations.

2. Terminology. You wanted to discuss this in a language that the average reader would understand, but quickly introduced the formal term: equivalence classes.I know what you mean, and I think this is a good term.

3. You argue that although different equivalence classes have different probabilities, the fact remains that the overall probability doesn't change.

4. This is true! Consider a standard deck of cards. If you select 1 card, you will have a 1/52 chance of picking the Ace of Spades. But, if you were to gamble on this, you would know that you that there are 13/52 Spades, and 4/52 Aces, meaning you have a 25.0% chance of picking a spade, and a 7.69% chance of picking an Ace. Multiplying these together, you have a 1/52 chance of picking the Ace of Spades.

5. The concept of equivalence classes is well established by the lotteries. They quote the odds for getting the Jackpot Winner, and 5, 4, 3 ... etc. correct.

6. In fact, the Powerball tells us that our overall odds of having some type of winner is 1/36.6.

7. This means they think in terms of Winners and Losers. The probability of winning something is 2.73% and the probability of losing is 97.27%.

8. The lotteries are gambling on these probabilities, or equivalence classes. If randomness is correct, they will win, bringing in more money than they payout.

9. My post gives the lottery players more information about their odds. Will it insure that they win? No. But, it helps to make more educated decisions.

The lotteries don't care what number is picked, only the odds or probability of having to make payouts.

Therefore, giving players more information about equivalence classes helps them to identify a playing strategy that improves their odds winning the jackpot.

If a player bets on a class, and not individual numbers, then they may just win. Of course, there is a huge cost associated with playing classes. We need to figure out how to make this affordable.

Lastly, let me clarify the hi-low distribution for you. I used a sequence that begins with the number 1 and ends with the number 53. If you generate all the combinations that start with 1, and end with 53, you will get my numbers. From your question, the numbers (1,19,22,38,53) is in the set, and (1,19,22,38,54) is NOT. Check it out.

Once again, thanks for participating, and try to think of ways to capture this information into a meaningful winning strategy.

JL ...

(PS: I'm staying away from your comments about creating smaller buckets. Your discussion introduces Integral Calculus and limits, and I think we should stay away from this for now.)

5. This is a nice blog. I like it!

6. Have you tried to apply Chaos Theory to determine a pattern in your lottory information?

7. 8. 