Still A, obviously, because there are way more mathematicians than mathematicians who are also plumbers. Nothing about your rephrasing nullifies the fact of the vastly differing base rates.
The only way you could get B to be correct, is if you stipulated that there was an equal (or, at any rate, a greatly distorted) chance of either group being picked for X to be drawn from it. But ignorance of this fact—that such a stipulation is needed, and that otherwise B cannot be correct—is precisely the fallacy!
Edit: Did you have in mind the additional premise that only plumbers can fix toilets?
“Conjunction fallacy is not actually a fallacy” seems like overselling the result, but Dr. Jamchie has a point. Imagine I’m thinking of a person named Bob. You have two hypotheses:
1) “Bob is a mathematician” with probability 0.5
2) “Bob is a mathematician and a plumber” with probability 0.1
Then you receive evidence that Bob fixed my water heater, updating the probabilities to 0.4 and 0.2. The first one is still higher, as it should be, but the second one got a bigger boost from the evidence (aka “likelihood”).
Still A, obviously, because there are way more mathematicians than mathematicians who are also plumbers. Nothing about your rephrasing nullifies the fact of the vastly differing base rates.
Let me give you an analogy: There are two bags of balls. 1st have 1000000 white balls and 10 black balls. 2nd have 5 black balls and no white balls. Bob took a ball from one of the bags and it was black. Which bag he took it from? There are more black balls in first bag, than in the second. As there are more mathematicians, who can fix water heater, that mathematicians who are plumbers. Still the correct answer would be 2nd bag obviously.
Did you have in mind the additional premise that only plumbers can fix toilets?
No, just that plumber have much higher probability of doing so.
″ In your analogy, no ball that is in the second bag is also in the first bag. However, all mathematician-plumbers are also mathematicians. ”
Most of balls in first bag are in fact not plumbers, as in real life, but whose who are—they are in the bag also. We could number the balls, and first 5 of 10 black balls in first bag would have same numbers as 5 balls in second bag.
1) 1000000 white balls and 10 black balls, numbered 1-10.
2) 5 black balls, numbered 1-5.
And now the question is: Bob drew a ball from a bag. Which is more likely?
1) It was a black ball with a number between 1 and 5.
2) It was a black ball with a number between 1 and 10.
~~~
I considered submitting the above as my full response, but here is another approach.
You seem to be substituting a question about the process of choosing for the original question, which was about outcomes. An example where your approach would actually be correct:
“We know that Alice has access to two lists online: an exhaustive list of mathematicians, and an exhaustive list of mathematician-plumbers. We know that Alice invited Bob over for dinner by choosing him from one of those two lists. We know that, by complete coincidence, Alice’s toilet broke while Bob was over. We know that Bob successfully fixed Alice’s toilet. Which list did Alice originally choose Bob from?”
In that case, it’s likely that the Bayesian calculation will say she probably used the Mathematician-Plumber list.
But notice that last question is different from the question of “which of the online lists is Bob most likely to be on?” We know that the answer to that is the Mathematicians list, because he has a 100% chance of being on that list, where he only has a high-probability chance of being on the Mathematician-Plumbers list.
Yes, but you see now, with enought details added, second question doesn`t seem to make a lot of sense. “Which” in the question implies that Bob is just on one of the lists, but most likely he isn’t. That being said, natural language does not correspond 1:1 to math or statistics. Some ambiguities are expected and a lot of sentences are up for interpretation. Now who is to say that second question you prodived is the correct way to interpret the original problem, and first one is not? First is at least coherent, while second is condradicting itself.
Still A, obviously, because there are way more mathematicians than mathematicians who are also plumbers. Nothing about your rephrasing nullifies the fact of the vastly differing base rates.
The only way you could get B to be correct, is if you stipulated that there was an equal (or, at any rate, a greatly distorted) chance of either group being picked for X to be drawn from it. But ignorance of this fact—that such a stipulation is needed, and that otherwise B cannot be correct—is precisely the fallacy!
Edit: Did you have in mind the additional premise that only plumbers can fix toilets?
“Conjunction fallacy is not actually a fallacy” seems like overselling the result, but Dr. Jamchie has a point. Imagine I’m thinking of a person named Bob. You have two hypotheses:
1) “Bob is a mathematician” with probability 0.5
2) “Bob is a mathematician and a plumber” with probability 0.1
Then you receive evidence that Bob fixed my water heater, updating the probabilities to 0.4 and 0.2. The first one is still higher, as it should be, but the second one got a bigger boost from the evidence (aka “likelihood”).
Let me give you an analogy: There are two bags of balls. 1st have 1000000 white balls and 10 black balls. 2nd have 5 black balls and no white balls. Bob took a ball from one of the bags and it was black. Which bag he took it from? There are more black balls in first bag, than in the second. As there are more mathematicians, who can fix water heater, that mathematicians who are plumbers. Still the correct answer would be 2nd bag obviously.
No, just that plumber have much higher probability of doing so.
This seems to be a problem of partitioning.
In your analogy, no ball that is in the second bag is also in the first bag. However, all mathematician-plumbers are also mathematicians.
In other words, your analogy is comparing these options:
1) Bob is a mathematician and Bob is not a plumber.
2) Bob is a mathematician and Bob is a plumber.
In that comparison, it is indeed possible that #2 is more likely.
But the actual problem asks you to compare these options:
1) Bob is a mathematician, and Bob either is a plumber or is not a plumber.
2) Bob is a mathematician, and Bob is a plumber.
Since all Bobs in #2 are also in #1, #2 cannot be more likely than #1.
″ In your analogy, no ball that is in the second bag is also in the first bag. However, all mathematician-plumbers are also mathematicians. ”
Most of balls in first bag are in fact not plumbers, as in real life, but whose who are—they are in the bag also. We could number the balls, and first 5 of 10 black balls in first bag would have same numbers as 5 balls in second bag.
Sure, so now there are two bags:
1) 1000000 white balls and 10 black balls, numbered 1-10.
2) 5 black balls, numbered 1-5.
And now the question is: Bob drew a ball from a bag. Which is more likely?
1) It was a black ball with a number between 1 and 5.
2) It was a black ball with a number between 1 and 10.
~~~
I considered submitting the above as my full response, but here is another approach.
You seem to be substituting a question about the process of choosing for the original question, which was about outcomes. An example where your approach would actually be correct:
“We know that Alice has access to two lists online: an exhaustive list of mathematicians, and an exhaustive list of mathematician-plumbers. We know that Alice invited Bob over for dinner by choosing him from one of those two lists. We know that, by complete coincidence, Alice’s toilet broke while Bob was over. We know that Bob successfully fixed Alice’s toilet. Which list did Alice originally choose Bob from?”
In that case, it’s likely that the Bayesian calculation will say she probably used the Mathematician-Plumber list.
But notice that last question is different from the question of “which of the online lists is Bob most likely to be on?” We know that the answer to that is the Mathematicians list, because he has a 100% chance of being on that list, where he only has a high-probability chance of being on the Mathematician-Plumbers list.
Yes, but you see now, with enought details added, second question doesn`t seem to make a lot of sense. “Which” in the question implies that Bob is just on one of the lists, but most likely he isn’t. That being said, natural language does not correspond 1:1 to math or statistics. Some ambiguities are expected and a lot of sentences are up for interpretation. Now who is to say that second question you prodived is the correct way to interpret the original problem, and first one is not? First is at least coherent, while second is condradicting itself.