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Thread: Why does the expected outcome matter? (statistics)

  1. #1 Why does the expected outcome matter? (statistics) 
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    You know how if you have 25% chance to win 20$, this is equivalent to a 100% chance of winning 5$ (it uses the assumption that 20$ are four times as important as 5$, which isn't 100% true and it probably would be much less true if we use 1 in millions chance to win millions of dollars vs. 100% chance to win 1$). I'm having some trouble with proving why is this approach the best approach to probabilities. I just know that it is and I have evidence that is really hard to explain.

    I'm trying to explain to my father why he shouldn't play the lottery, but I don't know how to explain it to them. He is highly intelligent, but he isn't good with counter-intuitive things (a.k.a. dysrationalia, if you're into cognitive psychology).

    The above text is my own hypothesis, trying to prove the expected outcome approach - if it's vague (it probably is), just ignore it, it's not vital for my question.

    One of the explanations I thought of is that if there is 50% chance to win something, that would mean there are two alternative "universes" (the only difference between them is that in one of them, I won that thing) and they are equally probable, so I might as well count that they both semi-happened (can I? this is the part of my hypothesis that might contain a fallacy), because that would be like they both happened, but then I divide it by two - and by "semi-happened", I mean that everything counts to a half-degree (so most things would remain the same if it was a 50% chance to win 1$). There is a single result (which is the average of the two possible results - they don't have to be two), which is what I need - when I consider a decision which has probabilities (say, playing the lottery), I'll compare two results - in one of them, I take the result I got from combining the probabilities, and in the other one, I take the result I'll get when I decide to not make the decision. The second result is often me winning/losing 0$, so it all depends on whether I have a profit or loss at the first result. That is how I deal with situations with probabilities, I just need a way to explain this approach understandably, because I sometimes have issues with that. Any alternative explanation would be very useful, this is probably what am I looking for, because my explanation doesn't seem convincing enough and is very counter-intuitive and hard to understand (even if I find a way to explain it more clearly). It doesn't have to be clear, though - I'm also looking for a way to explain it to myself, because I'm not certain it's true at all - there should probably be a widely accepted explanation, which is what I'm hoping to get.

    Also, any criticisms of my ideas are welcome! I don't expect you to agree with me, and if you prove me wrong, that would help me.


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  3. #2  
    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by Blackened View Post
    You know how if you have 25% chance to win 20$, this is equivalent to a 100% chance of winning 5$
    No it's not. If that was true then you would always win something in any game of chance. Obviously that isn't the case.


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    Quote Originally Posted by Markus Hanke View Post
    No it's not. If that was true then you would always win something in any game of chance. Obviously that isn't the case.
    I mean it's equivalent in terms of efficiency - it's obviously not equal, and I probably misused the word "equivalent". I meant to say that after making the assumption I mentioned above, in terms of decision-making, it doesn't matter whether you choose 25% to win 20$ or 100% to win 5$.
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    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by Blackened View Post
    You know how if you have 25% chance to win 20$, this is equivalent to a 100% chance of winning 5$
    No it's not. If that was true then you would always win something in any game of chance. Obviously that isn't the case.
    He means the expected value, the probability multiplied by the winnings, what you would expect to win if you played the game over a period of time, are the same.

    If you played a free game with no entrance fee:

    .25 * $20 = $5
    1 * $5 = $5
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    And that's why Blackened's father shouldn't play the lottery, the ticket cost paid by many is set to outweigh the winnings of the few. In most cases the accumulated money spent will also outweigh the winnings of an individual, since the chances of winning big are so very small.
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  7. #6  
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    Quote Originally Posted by wallaby View Post
    And that's why Blackened's father shouldn't play the lottery, the ticket cost paid by many is set to outweigh the winnings of the few. In most cases the accumulated money spent will also outweigh the winnings of an individual, since the chances of winning big are so very small.
    This isn't good enough, he'll simply say that he might be one of the few who win a lot, especially if he wins the jackpot (completely disregarding the base rate). I want explanation to why everything can be converted to its expected outcome (or if it can't, then why).
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    I want explanation to why everything can be converted to its expected outcome (or if it can't, then why).

    An expected outcome (a mean) isn't a prediction of if you'll succeed in a trial or not. It is a predicition of how many trials will pan out. Your father is correct in saying that knowing the expected value doesn't really apply to him. For example, you might run a restaurant and notice that you're selling 70% beef to 30% chicken. You can buy accordingly. If your father sits down and orders, you have no idea whether he's going to select beef or chicken.
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    I don't have to know whether he'd going to select beef or chicken - I never claimed that the expected outcome says that. It just combines information - for this case it can be still applied, but it will be vague and depending on many variables (most of which would remain unknown in most real life situations, so it won't be very useful). So, for this case, I'll be able to predict the general result of him coming to my restaurant - the average of the possible outcomes. The claim I'm trying to prove (or disprove) is that A is equivalent to B:
    A. He comes to my restaurant.
    B. The expected outcome of he, coming to my restaurant.

    They are obviously not the same thing, but I'm trying to say that they are equivalent.
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  10. #9  
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    Quote Originally Posted by Blackened View Post
    This isn't good enough, he'll simply say that he might be one of the few who win a lot, especially if he wins the jackpot (completely disregarding the base rate). I want explanation to why everything can be converted to its expected outcome (or if it can't, then why).
    Well he is right, he may well win the jackpot. (try convincing him then) If the odds of winning are one in a million, which are ridiculously good odds compared to the actual lottery odds, and he plays every week then we expect him to win once in every 19,231 years. However this win could come on any given week in that time interval, i say could because even if he plays for 19,231 years it's still not definite that he will win.

    As another example, one website claims that the odds of my car being stolen in a year are 1 in 159. So i would expect that after 159 years my car will have been stolen at least once. Now consider that as i'll probably only own a car for less than 65 years, should i still insure my car against theft, or am i wasting my money?

    What i'm saying is that the expected outcome only applies to a statistically large sample of either attempts or people.
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    Quote Originally Posted by Blackened View Post
    Quote Originally Posted by Markus Hanke View Post
    No it's not. If that was true then you would always win something in any game of chance. Obviously that isn't the case.
    I mean it's equivalent in terms of efficiency - it's obviously not equal, and I probably misused the word "equivalent". I meant to say that after making the assumption I mentioned above, in terms of decision-making, it doesn't matter whether you choose 25% to win 20$ or 100% to win 5$.
    Ok, I get it now.
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  12. #11  
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    Quote Originally Posted by wallaby View Post
    Quote Originally Posted by Blackened View Post
    This isn't good enough, he'll simply say that he might be one of the few who win a lot, especially if he wins the jackpot (completely disregarding the base rate). I want explanation to why everything can be converted to its expected outcome (or if it can't, then why).
    Well he is right, he may well win the jackpot. (try convincing him then) If the odds of winning are one in a million, which are ridiculously good odds compared to the actual lottery odds, and he plays every week then we expect him to win once in every 19,231 years. However this win could come on any given week in that time interval, i say could because even if he plays for 19,231 years it's still not definite that he will win.

    As another example, one website claims that the odds of my car being stolen in a year are 1 in 159. So i would expect that after 159 years my car will have been stolen at least once. Now consider that as i'll probably only own a car for less than 65 years, should i still insure my car against theft, or am i wasting my money?

    What i'm saying is that the expected outcome only applies to a statistically large sample of either attempts or people.
    Of course, but if you put it like that, he might win even after playing more than 19,231 years, or more importantly, he might even win today - you can't deny that. That's his whole point. The point I'm trying to prove is that every time he plays the lottery, if the ticket is 2$, then that's equivalent to losing something like 1.50$ (or even more), because of the expected outcome - in other words, that his small chance of winning the jackpot is equivalent to 100% chance of winning something like 0.10$. So it's like he's giving them money for nothing. I'm surprised nobody yet heard of any theory that proves this - I thought it was popular, because it's obviously very useful (I use it for many things).
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  13. #12  
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    Quote Originally Posted by Blackened View Post
    Of course, but if you put it like that, he might win even after playing more than 19,231 years, or more importantly, he might even win today - you can't deny that. That's his whole point.
    It's called gambling, of course the odds are stacked so that the company administering the lotto makes a profit. It does make for a very bad investment for exactly the reason you provided. But as with all gambling the issue is not with the expected return after 19231 years of play, it's with the allure of millions of dollars paid out for what the gambler hopes is a smaller amount. I found this article that may be of interest to you, although it deals with gambling in general it does raise some interesting points on how lotto companies get people playing.

    As for the mathematics, this may be of interest to you.
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