A Lottery Paradox
Gail Howard, who sells lottery winning systems, says "Six consecutive numbers (such as 1-2-3-4-5-6) have never been drawn in any state or international lotto game. It is highly unlikely that, if it has never happened before, it will happen this week just because you put your money on it! Our common sense lottery formula tells you not to play six consecutive numbers on one game panel."
Common sense can be unreliable; let's calculate. There are COMBIN(56,6) = 32,468,436 ways of choosing six numbers from the range 1 to 56. Subtract 51 ways of choosing combinations of numbers all consecutive and that leaves 32,468,385 combinations whose numbers are not all consecutive. This suggests a probability of 0.000001571 that the drawn numbers will be all consecutive and a probability of 0.999998429 that they will not. Could Gail Howard be right?
Edit: Check out this article on a "Fantasy 5" consecutive number draw in Florida back in 2011. Instead of one or two people splitting the pot for hundreds of thousands each, 47 people picked the consecutive numbers that hit, resulting in a 47 way split for a whopping $4k each. Avoiding a massive split is a better reason for not picking consecutive numbers than the fallacy that they never hit.
You can get an idea of how many lotteries Gail has studied by visiting her web site. The URL will not reproduce here but it is based on the word "smartluck."
I think the answer would be "all of them, foreign and domestic." She may be right about a 6-number sequence never having come up but I think a couple of sequences have come up in 5-number lotteries.
Elsewhere on her site she warns about not playing the numbers 1, 2, 3, 4, 5, 6 because it is a popular combination and if it won the prize would have to be shared with many other players. This is good advice, but what I am asking is should you not play sequences because they are rarely drawn?
Gail's first error is in believing that a combination's coming up or failing to come up affects its appearance in the future. One place on her site she says "Never play a combination that has just come up because it will be thousands of spins before it is due to come up again."
Gail is a fountain of fallacies. While she quotes dozens of customers who have won jackpots using her systems, she never has won one herself.
Quote: kratchikIn a lottery you select six numbers in the range 1 to 56. If the numbers are chosen randomly, it is generally believed that all possible combinations of these numbers are equally likely? If you agree then you are ready for the paradox.
Gail Howard, who sells lottery winning systems, says "Six consecutive numbers (such as 1-2-3-4-5-6) have never been drawn in any state or international lotto game. It is highly unlikely that, if it has never happened before, it will happen this week just because you put your money on it! Our common sense lottery formula tells you not to play six consecutive numbers on one game panel."
Common sense can be unreliable; let's calculate. There are COMBIN(56,6) = 32,468,436 ways of choosing six numbers from the range 1 to 56. Subtract 51 ways of choosing combinations of numbers all consecutive and that leaves 32,468,385 combinations whose numbers are not all consecutive. This suggests a probability of 0.000001571 that the drawn numbers will be all consecutive and a probability of 0.999998429 that they will not. Could Gail Howard be right?
0.000001571 + 0.999998429 = 1. As expected.
51 ways of selecting consecutive numbers : 32,468,385 ways of selecting non-consecutive numbers
That seems all pretty rational so far.
1 in 32,468,436* chance that your set of numbers will be picked whether consecutive or not.
I see no paradox except that picking any set of numbers will not pay back anywhere near 32,468,436 times the stake because the house edge will be of the order of 25% or more.
So. Don't do lotteries: Simple system.
Choosing a set that fits into any 'obvious sequence' like 1,2,3,4,5,6 is very likely to result in lots of sharing of your prize.
* I trusted your maths, probably unwisely.
The state's take on state lotteries is around 50%.
Quote: kratchikOnceDear: The question is "Could Gail Howard be right?" The paradox is, if she is, she contradicts a generally accepted fact about lottery number combinations. .
Could she be correct in what assertion?
Quote: kratchikGail Howard, . . . says "Six consecutive numbers (such as 1-2-3-4-5-6) have never been drawn in any state or international lotto game. It is highly unlikely that, if it has never happened before, it will happen this week."
Something incredibly unlikely has never happened, so don't bet on such incredibly unlikely events.
I've never won the lottery, so on past form, I should never play again.
Sorry. I still don't see any paradox.
No, the state take is usually fifty percent for education (ie, teacher salaries) and an additional fifteen to twenty percent for 'administration and advertising'.Quote: kratchikThe state's take on state lotteries is around 50%.
It's not fraud unless the state says it's fraud. They regulate themselves. Belie dat!Quote: kratchikonenickelmiracle: Fraud would not be tolerated in a state-run lottery. You are talking about slot machines. Today's machines are phony. The wheels don't mean a thing. They just stop where the computer chip tells them to.
Respect my athorita
I suspect this is a bad set of numbers.
When I play keno, I choose numbers 1 to 4 or 1 to 7 because they have the same chance of any other numbers. I suspect it also drives other players mad.
Those are pretty popular numbers you play for keno.Quote: CrystalMathCan you please research the number of times these numbers have come up: 24 12 17 14 4 37 ?
I suspect this is a bad set of numbers.
When I play keno, I choose numbers 1 to 4 or 1 to 7 because they have the same chance of any other numbers. I suspect it also drives other players mad.
Quote: onenickelmiracleThose are pretty popular numbers you play for keno.
Would be a shame if he had to chop with others once he finally hit that 7/7.
You will respect my authoritah.Quote: RSWould be a shame if he had to chop with others once he finally hit that 7/7.
Is Gail right or wrong? If she is wrong, what has she overlooked? I have been trying to find a clear and elegant explanation. Perhaps you can provide one.
Gail has many of these. I would love to try to straighten her out but I can't find a way to approach her.
Here's a real paradox for you:
There is one barber in Tippington and he shaves only the men who do not shave themselves.
Get it?
Quote: RSA paradox is (essentially) where a statement is both true and false at the same time. I'm not seeing a paradox here
It's true according to Gail and false according to some other people. I quote a better one in my reply to the Wizard.
Quote: kratchikIt's true according to Gail and false according to some other people. I quote a better one in my reply to the Wizard.
Then either one or the other are incorrect. But that's not a paradox.
Is this a paradox: I think 2+2=5 is true and you think it's false. ?
Quote: kratchikLet's just ask whether Gail Howard is right or wrong. She has examined thousands of lottery drawings and has failed to find a numerical sequence in a 6-ball drawing. She concludes that such a sequence is extremely unlikely.
She is right that such a sequence is extremely unlikely. That's as far as it goes.
If she took ANY OTHER 55 sets of numbers, she could then go and examine thousands of lottery drawings and would also fail to find those numbers to have been drawn.
Which part of "Each set of 6 numbers is equally unlikely" doesn't she grasp?
Which part of "What numbers have been drawn in the past has no relationship to what will be drawn in the future" can't she understand?
Quote:Therefore, she concludes, you should never play a sequence
Strike the words 'therefore' and 'a sequence' and she has the fair assessment
Quote:'she concludes, you should never play
Other people are right. Why would anyone 'crunched some numbers that Gail might use in an attempt to support her position.' Her position is untenable. If she wants to prove it, that is her job. Hint: You won't be able to prove it or support it.Quote:... Other people say that this is nonsense, that all combinations are equally likely.
Quote:Gail has many of these. I would love to try to straighten her out but I can't find a way to approach her.
Save your breath. She has a vested interest in not believing you.
Oh, you mean that famous barber Beardy McBeardyQuote:Here's a real paradox for you:
There is one barber in Tippington and he shaves all the men who do not shave themselves. Who shaves him?
Quote: kratchikPerhaps I used an inappropriate title. Let's just ask whether Gail Howard is right or wrong. She has examined thousands of lottery drawings and has failed to find a numerical sequence in a 6-ball drawing. She concludes that such a sequence is extremely unlikely. Therefore, she concludes, you should never play a sequence but instead choose numbers that are not in sequence. Other people say that this is nonsense, that all combinations are equally likely. I crunched some numbers that Gail might use in an attempt to support her position.
Is Gail right or wrong? If she is wrong, what has she overlooked? I have been trying to find a clear and elegant explanation. Perhaps you can provide one.
What she overlooked is surveying billions of more lottery outcomes, which don't exist. However, she could easily simulate lottery outcomes and it would show every set of picks wins roughly the same number of times, give or take for random variation. This would be like me picking a card from a deck of cards, with replacement, ten times. I note that ace of spades was picked so conclude that its probability is less than the other cards.
This is no way deserves to be called a paradox. It is can be called an example of a terrible use of statistics.
Quote:There is one barber in Tippington and he shaves only the men who do not shave themselves.
That's not how you tell it. The proper wording is, "The barber is the one who shaves all those, and those only, who do not shave themselves. The question is, does the barber shave himself?"
I don't consider this a paradox, or a good paradox, because such a person can't exist. It is a contradiction in terms. One I like better of this sort, goes like this:
It is given that god can do anything. The question is, can god create a rock so heavy that god himself cannot lift it?
At the risk of changing the topic, here is one I've brought up here before and have yet to get a satisfactory answer from anybody but myself.
Answer the following questions?
1. What is the probability that a random number, chosen from the uniform distribution between 0 and 10, will equal pi?
2. Can a random number, chosen from the uniform distribution between 0 and 10, equal pi?
I submit the answers are 0 and yes. The question is, how can something with probability zero happen?
I agree that the answers to 1) and 2) are 0 and "yes". You could substitute the integer 4 for pi and have the same "paradox."Quote: WizardThe question is, how can something with probability zero happen?
In the uniform (continuous) distribution between 0 and 10, there is an infinite quantity of possible numbers that could be chosen, each of which has a probability of zero of being the random choice. In this case, infinity times zero equals 1; i.e., the expected quantity of numbers that were chosen.
Is that a valid explanation?
Quote: DocI agree that the answers to 1) and 2) are 0 and "yes". You could substitute the integer 4 for pi and have the same "paradox."
In the uniform (continuous) distribution between 0 and 10, there is an infinite quantity of possible numbers that could be chosen, each of which has a probability of zero of being the random choice. In this case, infinity times zero equals 1; i.e., the expected quantity of numbers that were chosen.
Is that a valid explanation?
You're right that choosing 4 would be equally valid, but I don't think would make the point as well. I see what you're saying. However, we've all been taught around 4th grade that anything multiplied by zero is zero. It seems logical, because there is nothing to multiply.
Maybe a better way to say the probability of picking pi approaches zero.
I think the challenge there is that saying something "approaches zero" is usually accompanied by saying "as ...", meaning that there is some other independent variable approaching some particular value: perhaps something like "as the number of equally-likely possibilities approaches infinity, the probability of each one of those approaches zero."Quote: WizardMaybe a better way to say the probability of picking pi approaches zero.
In your example, though, we don't have a increasing number of possible choices; it is always the infinity of possible choices between 0 and 10.
I guess it's obvious I'm not a mathematician.
Are there are more, less, or the exact same amount of numbers between 0 and 10 compared to between 0 and 100.
Gail is in the business of selling publications and software for winning the lottery and on her web site she gives many "free tips" like this one. There are a number of books on winning lotteries and while some are scams, some are perfectly sincere. Richard Lustig, the author of the best-selling book on the subject, has confused luck with system, in which he sincerely believes. I believe that Gail also is sincere because while she claims that many of her customers have won lotteries, she admits never having won one herself. I hate seeing a good mind going down the wrong path and I am seeking convincing rebuttals for this tip and many of her others, ones that Gail could understand if they were presented to her.
No! Stoppit. If she wants to spout loblocks, let her. If you really think the untrained majority of readers are so uneducated, then lobby your education authorities to do better.Quote: kratchik... to solicit a clear explanation of why a combination that is in sequence is just as likely to be drawn as one that is not in sequence despite the fact that sequences are rarely drawn. This goes against Gail's "common sense," and I think the untrained majority of readers would agree with her.
Then regardless of her apparent sincerity, she's wrong. Oh so terribly wrong. You should be calling her out, deriding her systems and wising up her chumps. Don't publicise her name or her venture. Not as if she will listen, she's been spouting such nonsense for decades and no doubt grown rich from doing so. Round here we have a name for folks that spout nonsense for profit. It's not a nice name.Quote:Gail is in the business of selling publications and software for winning the lottery
Justify those absurd hypothesese.Quote:...some are perfectly sincere. . I hate seeing a good mind going down the wrong path.
Quote: OnceDearJustify those absurd hypothesese.
My opinions are the result of investigation. For details read "A Players' Guide to Lotto Strategies," which is available on Amazon in paperback and as an e-book. There it is explained that Richard Lustig's win record is mostly due to his daily buying of tickets and his playing the Florida Fantasy Five, which is an easy-to-win lottery. He is a likable fellow and is willing to talk to people on the telephone to explain his system. I had a long conversation with him and ended with "Good luck in the next drawing." He replied "Luck has nothing to do with it."
My faith in Gail Howard's sincerity stems from recognizing the extensive work she has done in compiling the rules of lotteries all over the world, examining tens of thousands of drawing results, and compiling statistics from them. Scam artists do not go to such trouble.
If you disagree with Gail's advice, and I believe you do, please give a good explanation that could be used in a classroom.
I refuse to be trolled or shilled. Goodbye.Quote: kratchikIf you disagree with Gail's advice, and I believe you do, please give a good explanation that could be used in a classroom.
Quote: OnceDearI refuse to be trolled or shilled. Goodbye.
Sounds like DorothyGale. Anyone remember her? Some of her sweet words to me were "You're a fine piece of work!" and "You couldn't program your way out of a paper bag." At least she knew some math and was a competent programmer.
Quote: kratchikSounds like DorothyGale. Anyone remember her? Some of her sweet words to me were "You're a fine piece of work!" and "You couldn't program your way out of a paper bag." At least she knew some math and was a competent programmer.
Well, and not that I care, but since the name your using here is new, are you saying that you've used another screen name?
