Statistics 101

I'm really not sure of the reasoning behind the statement. The a priori odds do not change based on the outcome. So the only sensible argument I can see is that if you win first time then the game must be rigged somehow. But even that is wrong. It's not AS wrong - but a single win is very unlikely to be sufficient evidence for such a conclusion.

Well Jar says you're being silly, so it's good that you agree.And he's right - your argument is silly. YOu've given no reason to believe that that person won for any different reason than other people win. Some number comes up, and one of the many people who play happens to have chosen it. It happens all the time.

I think that Modulus means that the probability that you have won given that you have the winning ticket is 1 (i.e. it is a conditional probability and not a very interestign one). The prior probability - the odds against winning without taking the draw into consideration - remain unchanged.

Modulous is right. It's all about conditional versus unconditional probabilities.In the Monty Hall example it is assumed that Monty intentionally picks a losing door. Since he can do that no matter which door you chose it doesn't affect the probability. If Monty chose a door at random, and it was a losing door that WOULD affect the probability that the door you chose was a winner - it would rise to 0.5.

No, Modulous is talking about both. First the unconditional prior probability, and then the conditional probability given that you know the numbers drawn match yours. If you're not seeing that then it's no wonder you're confused.

There is an easy way to work it out - although the problem is difficult enough that it's not going to be immediately obvious.There are three possibilities.You choose the right door (p = 1/3) and Monty choses a losing door (p = 1) The probability of that is 1/3 . 1 = 1/3You choose a wrong door (p = 2/3) and Monty chooses the other losing door (p = 0.5). The probability of that is 2/3 . 1/2 = 1/3You choose the wrong door and Monty chooses the winning door which also has probability 1/3.When you see Monty open a losing door the third possibility is eliminates so you divide the other two probabilities by the probability of the third option NOT occurring (1 - 1/3 = 2/3). And (1/3)/(2/3) = 1/2Or, even easie,r note that you are reduced to two equiprobable options and it follows from that that the probability is 0.5.Monty's behaviour is absolutely crucial to this problem. Dealing with different versions might be a good exercise for your class.

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I don't see how reading and comparing numbers is a function of probability. That's just a function of using your eyes.

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It's information relevant to the probability. Given that the numbers drawn match those on your ticket you know that you've won. It's that simple.

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Look, if I'm wrong, then there's somebody out there making a killing in the lottery - or at a casino, or somewhere - by having their friend read the numbers/cards to them instead of just looking at them directly.

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Only if your friend is precognitive. Casinos would go out of business if they let punters bet knowing the outcome. That's why they don't allow it. Is your friend precognitive ? If not how does he know the outcome at a time when you can still place a bet or buy a ticket ?

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Too simple, in fact, to be relevant to probability

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It obviously is relevant to the probability. Any informaton that lets you make a better guess at the outcome is relevant to probability. And what could be more relevant than actually knowing the outcome ?

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I don't see where that's stipulated in Mod's example.

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Well that's the problem isn't it ? You don't understand what Modulous is saying.Look at Message 16 again. Do you see ? The Powerball result is in. The numbers have been drawn from the machine. The numbers that were actually drawn match those on the ticket. The probability that Modulous has won GIVEN THAT INFORMATION is very different from the probability of winning without it. That's the point. Message 21 is also predicated on knowing what the winning number is. Message 25
Emphasis mine. THe key fact is knowiing the outcome, yet again. Multiple confirmation that the ticket has won. Do you see ? How can you have any confirmation unless the outcome ff the draw is known ?It's absolutely obvious that Modulous is talking about reassessing the probability based on extra information - in this case knowledge of the winning number. Edited by PaulK, : No reason given.

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How many times to I have to refute this?

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Once would do if it were false. But it isn't. Don't forgget tthat for manyu of your posts you're completely misunderstanding wwhat Modulous is talking about.

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The odds of heads coming up on a fair toss of a fair coin is 1/2. That's always true, by definition, regardless of whether or not you pick the coin up and put it in your pocket and never flip it, ever. The actual outcome has absolutely nothing to do with the odds, because the actual outcome doesn't change the sample space.

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The prior probability is always 0.5. But if you have relevant information you can reasonably use a conditional probability - to use a more interesting example the modified Monty Hall problem I discussed with Chiroptera upthread.The outcome is relevant information. While it is rather trivial, the probability that the coin came up heads given that it came up heads is 1. Not 0.5. To say otherwise is to claim that the result can be changed after it happened - and that it WILL change half the time !

I'd disagree - neither is the truly relevant probability (which is not calculable).For instance the lottery is won, more often than not. And for most lotteries the total number of people who have played them is less than the number of current players. And each of them and to enter for a first time. So it isn't surprising that there are some first-time winners. It would be surprising if there weren't.Equally everyone gets some instances of good fortune. Nobody has absolutely everything go wrong. Thus it is not surprising that any particular person can claim that they've had some good luck.To look at the probabilities of the specific items of good luck happening to one person after the fact is an error. An example of confirmation bias. If something else good happened they'd use the probability of that. The relevant probability would have to take account of all the events that MIGHT have happened that would be considered good enough to use in such an argument. And obviously that is going to be higher - probably much higher - than the prior probability of the actual events.

The lottery ticket has the number that you chose (past tense) printed on it. That's what Crash means.And from the point of view of the probabilities it DOESN'T matter which number you choose. Whatever you choose the probability of winning is the same. That's because all the possible numbers have the same probability of being drawn.Here's a little exercise for you. What's the chance of you correctly guessing the outcome of a coin flip - heads or tails, assuming a fair coin. Show your working. Can you find a strategy that has a success rate that is better or worse than 0.5 ? If you are right and we have to take your strategy into account then you should be able to do that. I say that you can't find a sensible strategy that has any other probability, because you are wrong.

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Just because they didn't happen doesn't mean they were any less possible

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That's not the point. When we know that they DIDN'T happen, we knwo that they CAN'T happen and we can reassess the probability using that knowledge. Just as in the Monty Hall problems - both the original and the variant I mentioned - when we gain knowledge that lets us exclude one or more possibilities we can generate a new proability, conditional on that knowledge. While it is trivial in the case of knowing the result it can be useful - as it is in the Monty Hall examples.

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That doesn't make any sense

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Makes perfect sense to me.

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And that certainly bears absolutely no relationship to probability as I was taught, which made no distinction between events that happened in the past and events that would happen in the future.

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That isn't the correct distinction. The important issue is how much knowledge we have. Past events are rather easier to gain knowledge of.

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Because, again, probability isn't about seeing the future. It's about relationships of outcomes, and we don't clear the sample space just because one of the outcomes already happened

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You're still not getting it. If we have information that lets us eliminate possibilities (or any other information that affects the likelihood of some or all outcomes) we can use that to produce a conditional probability conditional on that knowledge.The standard Monty Hall problem depends on that. We know that the other door has a 2/3 probability of being the winner because we have eliminated a possibility. If you were right you should use a 1/3 probabiity and say that you shouldn't switch. But you know that is wrong.

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As cavediver said, probability is most relevant to multiple repeated trials. But how could you generate probability if, for every trial in the past, you had to cross out your results and put down "1"? Saying that past events have a probability of "1" just erases all the work you did trying to establish a sample space. What could be the possible use of that?

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Again you are missing the fact that we have two probabilities here - a prior probability which assumes no special knowledge and a conditional probability which does use knowledge.

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It's nonsense, Paulk. Surely you see that by now? Pascal invented probability just so that we wouldn't have to think about events this way. You and Mod and Ned want to return us to a time when we had no idea what was a good bet and what was not. That's idiotic!

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In the standard Monty Hall problem we know that switching IS a good bet because we make use of new information to recalculate the probability. You say that we can't do that. Guess who's wrong.If you ever had a winning lottery ticket would you throw it away because it is "nonsense" to say that it won ? Would you say that the odds of it being the winning are still way too high ? Or would you actually accept that it did win, that you do know that it won ? I think that you would agree that it had won, and join with the view you are calling nonsensical.

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Show me from a probability textbook where probability says that, after an event happens, we cross out the odds and write in "1".

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Sorry, I don't have to prove your strawman. If you have a decent probability textbook it will deal with conditional probabilities. It would have to be a very elementary work to miss out that.

Take the standard Monty Hall problem, with three doors, A B and C.At the beginning the probability of the prize being behind each door is 1/3, correct ?You choose Door A. Monty chooses Door B. Using that information what is the probability now that the prize is behind Door C ?
What is the probability that it is behind Door B ?
What is the probability that it is behind Door A ?On the basis of these probabilities should you stick with A, switch to B or switch to C ? Or does it not matter what you do ?If it does matter, why, if it is not that we can use the extra information Monty has given us to reasess the probabilities ?

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You're arguing a strawman here. I've already said that the winning ticket is the winning ticket. The odds that it won, though, are still 1 in 146 million - regardless of the fact that we knew that it won.

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No, I'm saying that you are contradicting yourself. The position you are calling nonsense is fundamentally the position that we CAN know that we have the winning ticket.