I'm stuck at the first continue statement. Doesn't that suck up one of your num_trials? This can never happen per the conditions laid out, but then you put num_trials in the denominator when returning the percent of times it was green.
A confusing probability question: Red and green balls in an urn
11–20 of 169 posts
Re: A confusing probability question: Red and green balls in an urn
#12I think this is less about Bayesian thinking and more about misinterpreting the question, as another poster mentions: https://twitter.com/farrwill/status/1751788706355392639
ie i think many of the More Likely to be Green people are doing the math of if you pull from an urn with n/100 odds of getting red, your second pull will have odds (n-1)/99, which is less than n/100 for all n except n=100. Which is obviously a different question than should you bet on Green or Red.
Which is obviously a different question than "would you bet on the next ball being red or green"
Re: A confusing probability question: Red and green balls in an urn
#13Earlier quoted context omitted.
Yes, that's exactly right. Paraphrasing what another poster wrote about this on Twitter: "You would rather go fishing in a lake where you just saw someone else catch a fish." Also, the logic continues as you draw more balls from the urn. If the second ball is ALSO red, then you have even more evidence suggesting that N was selected in such a way as to make red the overwhelmingly more likely choice. Thus the chance of…
To extend the fishing metaphor, the premise of ~100 balls is also important: If each lake could only sustain one (good) fish, you would actively avoid anywhere someone had already caught one from.
That's "what's the probability you've caught the world's meanest fish".
Re: A confusing probability question: Red and green balls in an urn
#14Yes, but which door has the goat behind it?
Re: A confusing probability question: Red and green balls in an urn
#15Uniform n is what drives a lot of the interesting properties here. Other distributions can behave differently. For example if balls were inserted into the urn with their colors determined independently by coin flips, then instead of n being uniform it would be binomial. In that case, observing the color of a previously drawn ball would tell you nothing about the next one.
Re: A confusing probability question: Red and green balls in an urn
#16Earlier quoted context omitted.
To extend the fishing metaphor, the premise of ~100 balls is also important: If each lake could only sustain one (good) fish, you would actively avoid anywhere someone had already caught one from.
I guess the counterpoint to this is as the number of balls goes from 100 to infinite, it becomes increasingly harder to predict, correct?
According to the original poster on Twitter:
"One cool thing is that this probability doesn’t change if you replace 100 in the statement of the problem with some other number (as long as it’s at least 2), as these arguments show."
Source: https://x.com/littmath/status/1752167731682492522?s=46
Re: A confusing probability question: Red and green balls in an urn
#17I think this is less about Bayesian thinking and more about misinterpreting the question, as another poster mentions: https://twitter.com/farrwill/status/1751788706355392639 ie i think many of the More Likely to be Green people are doing the math of if you pull from an urn with n/100 odds of getting red, your second pull will have odds (n-1)/99, which is less than n/100 for all n except n=100. Which is obviously a di…
I'd argue that it's not so much a difference of opinion than it is just a reasoning error given the question as stated. That's sort of the whole point of this post-- this is a case where "doing the math" in the expected way gives the wrong answer, because the state of the system is cast in stone (in terms of the ratio of red/green balls in the urn) when you first start out, so it's all about leveraging the information you get from your first choice. I think this sort of reasoning is actually surprisingly useful and widely applicable in everyday life, like in investing or diagnosing the likely cause of a software bug from incomplete information.
Re: A confusing probability question: Red and green balls in an urn
#18Change it to drawing from a normal distribution... the answer will change dramatically and get very close to what one would likely consider "common sense", ie that it remains close to 50/50. I think binomial gets it to 50/50.
Humans aren't wired for uniform distribution. 100 reds seems crazy unlikely compared to 50/50, yet this trick question makes those equally likely.
Re: A confusing probability question: Red and green balls in an urn
#19I think this is less about Bayesian thinking and more about misinterpreting the question, as another poster mentions: https://twitter.com/farrwill/status/1751788706355392639 ie i think many of the More Likely to be Green people are doing the math of if you pull from an urn with n/100 odds of getting red, your second pull will have odds (n-1)/99, which is less than n/100 for all n except n=100. Which is obviously a di…
I'd argue that it's not so much a difference of opinion than it is just a reasoning error given the question as stated. That's sort of the whole point of this post-- this is a case where "doing the math" in the expected way gives the wrong answer, because the state of the system is cast in stone (in terms of the ratio of red/green balls in the urn) when you first start out, so it's all about leveraging the informatio…
It's definitely a poorly worded question. "More likely" is ill-defined. More likely than what? Than the last draw? Than drawing the other color just on this pick?
Re: A confusing probability question: Red and green balls in an urn
#20It's as ill posed as the Monty Hall problem. The issue is in what it means for the first ball to picked randomly.
Would the question be valid if the first ball was green? If not, then it's equivalent to the standard answer to the Monty Hall problem. Analyze it as if the first ball was not picked randomly (i.e. Monty intentionally picked the wrong door).