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Oh my lord you guys that is not how probabilities work. You can’t look at what actually happened and then say “Well that was the chance.” If I flip a coin 20 ti

by ebbv 8y ago

Oh my lord you guys that is not how probabilities work. You can’t look at what actually happened and then say “Well that was the chance.” If I flip a coin 20 times and happen to end up with a 15:5 split that doesn’t mean the chance of heads is 3:1. Also the context of that 1 in 12 in the quote is important. They’re saying that was the chance on the first flight in retrospect. The shuttle and it’s supporting facilities and processes didn’t stay static, they improved over time.

craftyguy 8y ago
I didn't say it was a probability. Quite literally, if you look at how many people flew and how many of those died, 1 in 25 died. There was a 100% 'chance' of 1 in 25 dying on the shuttle because that's what actually happened. There's no probability there.
drb91 8y ago
Wait, so a ratio by itself is meaningless without showing how the ratio was derived? If only we could verify the 1/12!
whatshisface 8y ago
If statistics didn't converge to probabilities in the high-n limit then science would be impossible. In the 15:5 coin example, the conclusion is skewed by a massive prior that coins are fair. When we're talking about things like space shuttle launches, it's not reasonable to expect our understanding of the system to be so good that we can boldly forge through with estimates that disagree with results.
joshuamorton 8y ago
>Oh my lord you guys that is not how probabilities work. You can’t look at what actually happened and then say “Well that was the chance.” If I flip a coin 20 times and happen to end up with a 15:5 split that doesn’t mean the chance of heads is 3:1. Erm, over a large sample size, yes this is exactly how things work. Sure 15/5 may be too small, but 150/50 isn't. That's well after the point where its reasonable to believe that your coin is rigged. As the other user mentioned, if you don't have the prior that the coin is fair, a 15/5 outcome may (and does) indeed imply that heads are more likely than tails. In fact, 15/5 would imply a less than 1% chance of a fair coin, given uniform priors. (Beta[15, 5] -> 1st percentile is 50.175)
- pps43 8y ago
  But in this case the samples are small. If you flip a coin once and got tails, what is the probability of heads?
  - joshuamorton 8y ago
    That's not quite the same question as what I'm answering. I'm saying "given you flip a coin once and it comes up heads, what is the probability that its a fair coin" (or previously, given you flipped a coin 20 times, and it came up 15/5, what is the likelyhood it is a fair coin). And with a single flip, you can't actually answer that question (BetaDist[1, 0] is undefined). Now, once you've answered that question, you can integrate to get a probability that the next coin is heads, but with a single flip you can't even answer the first question. You need to have gotten heads and tails each at least once.
    - isostatic 8y ago
      Flip a coin 50 times and get 50 heads, it's almost impossible it's a fair coin. Does your BetaDist function work in that case?
      - joshuamorton 8y ago
        No, the beta distribution BetaDist[a, b] is only defined for a, b strictly > 0. When a or b is very large and the other is zero, you can treat it as 1 and get a decent approximation, but it'll be a an estimate that biases towards the center.
      - joshuamorton 8y ago
        I should also mention that if you're willing to make assumptions like a uniform prior, this totally works, but it can lead to some surprising results for small a, b. Notably a uniform prior is Beta[1, 1]. Beta[2,1] implies something like a 75% chance of the true mean being above .5, which seems a tad overzealous.
    - pps43 8y ago
grondilu 8y ago
> If I flip a coin 20 times and happen to end up with a 15:5 split that doesn’t mean the chance of heads is 3:1. Isn't it the best you can say, though? 20 is a small number but when it's all you have, there's nothing you can do about it. Of course we know the "real" probability is a 1/2 because we know how coin tosses work, but if we didn't know it and coin tosses were black boxes, we'd have to go with this 3:1 chance.
- hikarudo 8y ago
  The chance of heads being 3:1 is the maximum likelihood estimate, which indeed is the best we can do if we don't know anything else. But usually we do have an idea of reasonable values.
quickben 8y ago
I believe he is right though. You are considering future probability chance, and are right, but he is considering past average. In this case his factual outcome trumps your potential forecast.

by ebbv 8y ago

craftyguy 8y ago

I didn't say it was a probability. Quite literally, if you look at how many people flew and how many of those died, 1 in 25 died. There was a 100% 'chance' of 1 in 25 dying on the shuttle because that's what actually happened. There's no probability there.

drb91 8y ago

Wait, so a ratio by itself is meaningless without showing how the ratio was derived? If only we could verify the 1/12!

whatshisface 8y ago

If statistics didn't converge to probabilities in the high-n limit then science would be impossible. In the 15:5 coin example, the conclusion is skewed by a massive prior that coins are fair. When we're talking about things like space shuttle launches, it's not reasonable to expect our understanding of the system to be so good that we can boldly forge through with estimates that disagree with results.

joshuamorton 8y ago

>Oh my lord you guys that is not how probabilities work. You can’t look at what actually happened and then say “Well that was the chance.” If I flip a coin 20 times and happen to end up with a 15:5 split that doesn’t mean the chance of heads is 3:1. Erm, over a large sample size, yes this is exactly how things work. Sure 15/5 may be too small, but 150/50 isn't. That's well after the point where its reasonable to believe that your coin is rigged. As the other user mentioned, if you don't have the prior that the coin is fair, a 15/5 outcome may (and does) indeed imply that heads are more likely than tails. In fact, 15/5 would imply a less than 1% chance of a fair coin, given uniform priors. (Beta[15, 5] -> 1st percentile is 50.175)

pps43 8y ago
But in this case the samples are small. If you flip a coin once and got tails, what is the probability of heads?
- joshuamorton 8y ago
  That's not quite the same question as what I'm answering. I'm saying "given you flip a coin once and it comes up heads, what is the probability that its a fair coin" (or previously, given you flipped a coin 20 times, and it came up 15/5, what is the likelyhood it is a fair coin). And with a single flip, you can't actually answer that question (BetaDist[1, 0] is undefined). Now, once you've answered that question, you can integrate to get a probability that the next coin is heads, but with a single flip you can't even answer the first question. You need to have gotten heads and tails each at least once.
  - isostatic 8y ago
    Flip a coin 50 times and get 50 heads, it's almost impossible it's a fair coin. Does your BetaDist function work in that case?
    - joshuamorton 8y ago
      No, the beta distribution BetaDist[a, b] is only defined for a, b strictly > 0. When a or b is very large and the other is zero, you can treat it as 1 and get a decent approximation, but it'll be a an estimate that biases towards the center.
    - joshuamorton 8y ago
      I should also mention that if you're willing to make assumptions like a uniform prior, this totally works, but it can lead to some surprising results for small a, b. Notably a uniform prior is Beta[1, 1]. Beta[2,1] implies something like a 75% chance of the true mean being above .5, which seems a tad overzealous.
  - pps43 8y ago

grondilu 8y ago

> If I flip a coin 20 times and happen to end up with a 15:5 split that doesn’t mean the chance of heads is 3:1. Isn't it the best you can say, though? 20 is a small number but when it's all you have, there's nothing you can do about it. Of course we know the "real" probability is a 1/2 because we know how coin tosses work, but if we didn't know it and coin tosses were black boxes, we'd have to go with this 3:1 chance.

hikarudo 8y ago
The chance of heads being 3:1 is the maximum likelihood estimate, which indeed is the best we can do if we don't know anything else. But usually we do have an idea of reasonable values.

quickben 8y ago

I believe he is right though. You are considering future probability chance, and are right, but he is considering past average. In this case his factual outcome trumps your potential forecast.