51–60 of 169 posts
In this specific case, P(redOnKthSample) = (numberOfRedSamples + 1) / (totalNumberOfSamples + 2) = (1+1)/(1+2) = 66% red.
On the second draw, if the ball is green, then you get P(redOnThirdSample) = (1 + 1) / ( 2 + 2) = 50%
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More likely than the other option, given the information. Getting an accurate grasp on the information is the whole point of the question.
Why is that the question? Why not "more likely than the last draw" ?
Or I could just say "I think most people interpreted the tweet correctly, and it would be annoying if it was overly precise"
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That's definitely fair, and I assuming in a classroom setting you'd be very consistent and precise about what that means. But this is a twitter poll, and I doubt people are that consistent. Ultimately I'm just pushing back on the conclusion that this is because "Bayesian thinking is really foreign to people." I think if the question had instead been "would you bet on the urn having started out with more red balls, gr…
That question is quite a bit of handholding. A closer analogue would be “would you bet the next ball is red / green / they’re equally likely?”. Personally I’d word it as “The next ball is… more likely to be red than green / more likely to be green than red / equally likely to be red or green / I don’t know.”
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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?
I think the wording is also biasing the outcome. The next ball is more likely to be green than the previous ball , and a lot of people will regard that posterior probability statement as implicit. If the question had been phrased differently eg, 'what is the most likely color for the next ball selected' I suspect more people would have voted for equal likelihood.
It'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).
edit: After thinking about it some more, it is NOTHING like handing the bag to someone else and having them remove a red ball. The whole point to the first draw is to draw the ball randomly. I think my second statement is still true. But to your original question, if the first ball isn't red, then the question is invalid (and you can't throw the ball in and choose again)
Follow up question: > You are given an urn containing 100 balls; n of them are red, and 100-n are green, where n is chosen uniformly at random in [0, 100]. You select 99 balls at random and find that they are all red. What is the probability the last ball is red?
50%. With current knowledge N is either 100 or 99 with an equal probability.
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Why is that the question? Why not "more likely than the last draw" ?
If, hypothetically, the majority of words communicated from one homo sapiens to any other homo sapiens (majority weighted by the number of homo sapiens who will hear, read, or otherwise receive the word over the course of the word's existence) were designed to be precise and unambiguous to the extent that the mean number of homo sapiens who receive the word correctly interpret the exact meaning of the word intended b…
Try this book out, it's a fun read on how minor changes in words can make big differences in readability: https://www.amazon.com/Sense-Style-Thinking-Persons-Writing-...
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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?
How is "more likely"I'll defined? They are two options red or green. Are you more likely to draw red or are you more likely to draw green.. how is it poorly worded?