Statistics 101

Ha! You are right. It took me a half hour playing with equations and drawing pictures of doors before I finally figured this out.We are calculating the probability P(A|B) = P(A \cap B) / P(B), where A = door #1 is the winner and B = door #2 is a loser. In this case, P(A|B) = 1/2. But this isn't the Monty Hall problem. In the Monte Hall problem, B = the door shown is a loser, in which case (A \cap B) = A and P(B) = 1, and so P(A|B) = 1/3 still. (\cap is LaTeX for the set intersection symbol, the upside down U).Gah. You really do have to be careful how you set these things up! (As the students in my class are finding out on the homework).Man, this sounds like a good problem to bring up in class.

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Actually, if their god makes better pancakes, I'm totally switching sides. -- Charley the Australopithecine

This is probabilistic nonsense.The odds that you have a winning ticket are 1 in 146 million because that's the odds of matching the lottery numbers with your ticket. There's no "depending on whether or not you matched." That's nonsense. It doesn't matter whether the lottery drawing happens in the past or the future; the odds that it wins that drawing are 1 in 146 million. It doesn't matter whether or not you know it's a winning ticket or not - the odds that it wins are 1 in 146 million.The reason people throw away their losing lottery tickets is because they know they don't match the numbers; but the fact that the ticket is in the trash doesn't change the fact that there's a 1 in 146 million chance that it matched the numbers for that lottery drawing.

Fair enough, but the lottery is unconditional probability. There's no communication between lottery players and the lottery draw-er, there's nothing conditional about the numbers except that the first 5 are from the same pool of numbers, and the 6th is from a separate pool. And you don't have the opportunity to change your numbers.The numbers might as well be generated simulatenously; they do it the way they do for dramatic reasons.It doesn't change the odds of you winning. Your odds don't change simply because your friend reads you the numbers in random order, or compares them all at once, or writes a sophisticated computer program, or what have you.Regardless of how you find out the results, your odds of winning are 1 in 146 million.

We are talking about two different things, I think.I am talking about the probability of an event happening that has already happened. You seem to be talking about the probability before the event happens.I don't know what you mean when you say "probabilistic nonsense." To me, it is nonsense to talk about the probability of an event that has already happened. It either did or it didn't. Once again, lack of knowledge about the outcome of an event that has already happened doesn't mean that there is a probability of the event going one way or another.

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Those who would sacrifice an essential liberty for a temporary security will lose both, and deserve neither. -- Benjamin FranklinWe see monsters where science shows us windmills. -- Phat

No, I'm not. I'm talking about the probability of the event at all points in time.The probability doesn't change before the event, or after the event, or whenever (unless of course you change the conditions of the event, like rolling a dice instead of flipping a coin - you know what I mean). I can tell you what the odds of getting a heads result on a coin flip are (given a fair coin), regardless of when you plan to flip the coin. Regardless of when the event happened - the future, the past - I can tell you what the odds of the outcomes are. I mean - according to how probability works, the statement has no meaning. But that's clearly not true. I can tell you the probabilities surrounding a coin flip, even if you already flipped the coin. You can always figure probabilities after the fact, as long as you can determine the sample space.A fair coin is 1/2 heads, 1/2 tails. That's the same for every flip - whether or not the flip has happened yet. Even if you flip the coin and it comes up tails, and you're standing right there looking at it, there's only a 1/2 chance of the coin came up tails. It did, but that doesn't make it any more likely.The fact that only one outcome can actually happen doesn't change the probabilities of other outcomes. Probability isn't about telling you what's going to happen, or what did happen; it's about telling you about the relationship between desired outcomes and all outcomes.Edited by crashfrog, : No reason given.

Perhaps we are just arguing semantics.dictionary.com says this:

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4. Statistics.
a. the relative possibility that an event will occur, as expressed by the ratio of the number of actual occurrences to the total number of possible occurrences.
b. the relative frequency with which an event occurs or is likely to occur.

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Wikipedia says this:

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Probability is the extent to which something is likely to happen or be the case.

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Under the first usage from dictionary.com, I stand by what I said. It's meaningless to talk about the probability of something that has already happened. The relative possibility that I was born on an even-numbered day is 1, because there are no other possibilities. The relative possibility that my first grandchild will be born on an even-numbered day (assuming I will have one) is 1/2, because that event has not yet happened. Once any event has happened, the relative possibility of that event is 1 because there are no other possible occurrences.It seems to me that you are using the term more in accordance with the second usage from dictionary.com and the Wiki definition, although Wiki seems ambiguous and could be consistent with my usage. In that sense, you are correct that the relative frequencies of the different possibilities of random events do not change simply because one outcome has been determined.

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Those who would sacrifice an essential liberty for a temporary security will lose both, and deserve neither. -- Benjamin FranklinWe see monsters where science shows us windmills. -- Phat

Not quite. The odds that you have a winning ticket is 146 million to 1 given the information you have. Once you know what the numbers in the lottery where they are now either 1 or 0. Yes - if we know what the next card is.However we don't, which is why when you are in this position you have to weigh up the odds of you getting that Ace and the amount you stand to win against the odds of not getting the Ace and the amount you stand to lose...in the long term.Otherwise the we could say the chances of getting heads is either 1 or 0. It is true, but 1 or 0 is not a probability. It is two probabilities. Since we don't know which one is true we have to work out the odds of each one being true. With cards there are 52 - 5 (47) cards and there are 4 aces in the deck. Therefore the probability of you getting the ace is 4 in 47 or about 1 in 12. If you stand to win 12 times more than you will lose, you will break even with your bet.We don't know what is coming though, so whilst it is either there it isn't - that is meaningless in terms of probability.

Correct. You responded to me after I made what I had considered a rather obvious point - and decided to dispute it. It is not me who is wasting your time. Correct. But the chances of it being in B has increased. Correct.However, your odds change as new information comes in. Its a perfectly straightforward sentiment. When you know you already have five balls, you have a greater chance of now becoming a jackpot winner. I've never disagreed that the odds are 1 in 146 million of winning the lottery you describe. So you agree that getting more information can change the probabilities of you winning. Like knowing you have a winning ticket for example. No. I'm asking what are the chances of winning the lottery that has just been drawn when you are holding the losing ticket. 0 right? And if it is a winning ticket it is 1. It isn't controversial, and you needn't have spent so long disagreeing with me on it. WERE always. Now they are not 1 in 146 million because you are sure you have the winning ticket. The odds are much more in your favour now. That isn't at issue either. We are talking about after the fact calculations here. So if it landed heads, what is the probability that it landed heads?Hopefully you will answer 'nearly 1'.

I disagree.The probability, under the first definition from dictionary.com above, of a past event happening is either 0 or 1. It either happened or it didn't. It makes no sense to say that the odds are 146 million to 1 given the information you have. Your information doesn't change the outcome of a past event. Likewise, the mere fact that we don't know what the next card is doesn't affect the card.Certainly, gamblers use probability calculations all the time in determining whether to bet on a given outcome. And, given that the outcome is unknown, it's reasonable for them to do so. However, in the example of a deck of cards, no calculation or discussion of probability will change the identity of the next card to come.

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Those who would sacrifice an essential liberty for a temporary security will lose both, and deserve neither. -- Benjamin FranklinWe see monsters where science shows us windmills. -- Phat

Subbie,Suppose Mrs. Smith is one of a million people who have bought a lottery ticket. In the lottery, only one ticket is drawn, and a million dollars is paid out to the winner. The ticket drawn happens to be Mrs. Smith's.Mrs. Jones has also bought a ticket in another lottery, in which ten million people take part. In this lottery too, one ticket is drawn and a million dollars paid out to the winner. It so happens that Mrs. Jones' ticket wins her a million dollars too.In retrospect, how should we assess Mrs. Smith's luck in relation to Mrs. Jones'?We can say: "Well, Mrs. Smith's winning a million dollars has already happened, so the probability of that occurence is 1. Likewise, the probability of Mrs. Jones' winning a million dollars is also 1, because it has already happened. Therefore, both women were equally lucky. Besides, any statement about Mrs. Smith's and Mrs. Jones' luck in their respective lotteries, after the fact, is meaningless."Or we can say: "There were a million people buying a lottery ticket in Mrs. Smith's case, and ten million in Mrs. Jones' case, so Mrs. Jones' chance of winning a million dollars was ten times less than that of Mrs. Smith's. Therefore, Mrs Jones was ten times more lucky than Mrs. Smith."Which meaning of the word 'probability' has more utility, you think?Edited by Parasomnium, : No reason given.

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"Ignorance more frequently begets confidence than does knowledge: it is those who know little, not those who know much, who so positively assert that this or that problem will never be solved by science." - Charles Darwin.

I still don't see how it does. You don't ever have an opportunity to act on that information; the question of whether or not you're going to win happens when you pick your numbers and the lotto people pick the winning ones. There's never an opportunity to act on the information, unless you're cheating somehow - cheating, in fact, would be the only way to raise your odds above 1 in 146 million besides buying more tickets.And I know this is so because the only way anybody has ever figured out how to reliably "game" the lottery is to buy more tickets - to buy so many tickets, in fact, that you've purchased one of every combination of numbers. No one has ever increased their odds of winning the lottery by having their friend read the numbers to them, or TiVoing the drawing and watching it in reverse, or what have you.There's only one way to game the lottery and that's to cover all the possibilities. (I'm not a gambler - is that what they mean by "covering the spread"? I've heard that term.) If changing the way you learn about the lotto numbers increased your odds of winning as remarkably as you misrepresent it, we'd see people doing that as a strategy. It was 1 in 146 million, just like it was for everybody else. You boughts your ticket and you took your chances. You lost. That doesn't mean you had no possibility of winning; just that you didn't. The odds didn't change just because you lost. No, they're the same as they were before. It doesn't matter that the drawing happened in the past and now you know what the result is. It was 1 in 145 million.You're displaying the exact same fallacious thinking that causes people to credit God when they win the lottery - that, somehow, winning is so unlikely that if it happened to you, it had to happen to you; it couldn't have happened to anybody else. But that's nonsense. The other outcomes don't disappear from the sample space just because they didn't happen. That's not what probability is about.It's about the relationship of desired outcomes to all outcomes, and that doesn't change just because one of those outcomes is what actually happened.And, contrary to your assertion, changing the way you learn about the outcome doesn't change how many of those outcomes are desired. There's only one desired outcome in this case - the one where your numbers are the ones that match. Changing how you find out about the numbers after the fact doesn't increase the number of those possible outcomes that lead to a big cash payout for you. Heads is one outcome out of a sample space of two, so the probability is the same as it was before you flipped it - 1/2. How many different ways do I have to say this before it sticks?Look, prove me wrong, empirically. Buy a lottery ticket tonight, and then have your friend read you the numbers in whatever way you think will maximize your odds of winning. When you win, you can fly me out to Manchester and I'll buy us a pitcher of Newcastle and tell you I was wrong.

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The odds of winning the lotto can be a million to one, but if I win it on my first try, then those weren't my odds were they?

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....... Wow.... just..... wow.............................

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"A man can no more diminish God's glory by refusing to worship Him than a lunatic can put out the sun by scribbling the word, 'darkness' on the walls of his cell." -C.S. Lewis

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.

I don't see how reading and comparing numbers is a function of probability. That's just a function of using your eyes.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. But I'm familiar with a considerable number of ways to cheat games of chance - which I think should be completely legal, since they're already cheating you in return - and this, to my knowledge, is not one of them.

Hmmmmm, where to start....It's not possible to fully determine the probability of either woman winning because you didn't specify the odds of winning. You specified the number of tickets bought, but that doesn't have to have any relation to the probability of winning. Moreover, it's not possible to determine the probable winning payouts necessarily based on the number of tickets sold.I assume that you were trying to describe a situation where Mrs. Smith had a higher probability of winning than Mrs. Jones. In that situation, it would be accurate to say that Mrs. Jones was luckier than Mrs. Smith.As far as which meaning of the word has more utility, I suppose that all depends on what one is trying to say.

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Those who would sacrifice an essential liberty for a temporary security will lose both, and deserve neither. -- Benjamin FranklinWe see monsters where science shows us windmills. -- Phat