3 ms·

Ok, would someone explain #6, "The Broken Water Heater Problem" to me? As expressed, there's nothing to say that the person (who repaired the heater) could not

by rskar 14y ago

Ok, would someone explain #6, "The Broken Water Heater Problem" to me? As expressed, there's nothing to say that the person (who repaired the heater) could not be just a plumber, or perhaps just a handyman and neither an accountant or a plumber. Yet, as demonstrated, the person is given to be an accountant. Event A is "accountant and plumber". Event B is "accountant and not a plumber". Hence A unioned with B means "accountant"; or, if that's too general, "is an accountant and may or may not be a plumber too". In any case, both events A and B qualify the person to be an accountant. So what's the controversy? When it comes to water heaters, we're all accountants?

scarmig 14y ago
If I'm understanding it correctly, it's (ab)using the word "accountant" to mean "someone who can receive payments." So the question is really "Is this person someone who accepts money, or is this person someone who accepts money and is a licensed plumber?" If you interpret it literally, it uses the law of the excluded middle inappropriately and would be just as true if you replaced "accountant" with "President" or "extraterrestrial angel."
- rskar 14y ago
  Thanks for the reply, but I doubt that the issue surrounds (ab)use of the word "accountant". If you go to the attributed source (at http://www.reddit.com/r/math/comments/12n04d/counterintuitive_mathematical_results_accessible/c6wion3 http://www.reddit.com/r/math/comments/12n04d/counterintuitiv...), instead of water-heater/accountant/plumber it's car/teacher/mechanic. I'm more inclined to believe this is a (poor) illustration of the idea that a sub-set is likely to be smaller than the set it came from: the set of all accountants vs. the sub-set of all accountants who are also plumbers. In which case - duh! - P(accountant) > P(accountant and plumber). As long as one can indulge in the naivety of this pedagogical tale, that the narrator (the "I" who said "I went to a person") wasn't too particular on whom he hired for the job, then sure, why not?, 'twas likely the accountant that fixed it. In the real world, which has teachers and mechanics in addition to accountants and plumbers and other occupations, and where people may tend to be pickier on who they'll let near their water heater - leave alone pay someone to mess with it - would this conclusion of "accountant" still prevail necessarily?
hackinthebochs 14y ago
Your breakdown isn't quite right. Event A is "account and plumber" while Event B is "accountant" (leaving plumber status unconstrained). Thought of this way its easy to see that event B is completely contained in event A, therefore P(A) >= P(B).
- rskar 14y ago
  Nope, take a closer look, Event B is pictured like a crescent moon, labelled "People who are accountants and not plumbers", and described thus: "There's also the probability 'B' that the guy who fixed my water heater is only an accountant and not a plumber." Plumber status within B is clearly "Not a plumber".
  - hackinthebochs 14y ago
    You're certainly right about the details you mentioned. But they are not relevant to the problem. "Accountant and not a plumber" was not in the question at all. Lets make it more precise. X = is accountant, Y = is plumber. The question is which is greater: P(X) or P(X and Y). It should be clear that the event "X and Y" is completely contained within the event "X". To put it another way, P(X) = P(X and Y) + P(X and ~Y) (we are summing over all possible unconstrained events, in this case Y and ~Y). Hence, P(X) >= P(X and Y)
    - rskar 14y ago
      Yeah, I think I picked up on that regarding P(X) >= P(X and Y) (see my reply to scarmig). From what NhanH said, I think I now see where my issue is.
NhanH 14y ago
The way it was explained in the infographic wasn't clear. But what he meant is that between P(X = accountant) and P(X = accountant and plumber), the former is bigger. The fact that P(X = plumber) > P(X = accountant) is irrelevant, as we wasn't concerned with P(X= plumber) at all. The implication of this "problem" is an interesting human biased: there appears to be cases where we judge the more detailed explanation as more probably, while statistically speaking, it isn't. If a person is described as "being concerned with discrimination and social justice", we tend to judge P(X = activist and Y) as more probable than P( X = Y). This is detailed in "Thinking, fast and slow" by Kahneman - a wonderful book if anyone haven't read it.
- rskar 14y ago
  I think you've shown me the way here. My issue, I suppose, is the problem with word problems in math. Generally that problem is two-fold. Does a word problem sufficiently describe the situation? And do you sufficiently understand the situation it describes? I believe this to be at the core of all the supposed "controversial math problems". BTW, thank you for suggesting Kahneman's book, it is now on my reading list. I did take notice that a sub-set is likely to be smaller than the set it came from: the set of all accountants vs. the sub-set of all accountants who are also plumbers. So therefore P(accountant) > P(accountant and plumber). Unless P(accountant) be zero, or P(plumber|accountant) be one, in which case neither is greater. Reflecting on my own human biases, I think I ended up reading it as if this were a multiple-choice question, and I had to pick the "best" choice. In other words, it isn't enough that this person is an accountant, but that this person is also a plumber. Or put another way, if I had to wager on whether anyone would pay just any old accountant to go mess with their water heater, I would find it more surprising that the particular accountant in fact hired had no demonstrable plumbing skills. I'd imagine Kahneman could teach me more on this. My favorite word problem joke goes something like this. If there are three cans on a fence, and you hit one with a thrown rock, how many cans remain on the fence? Answer: Two. Now, if there are three birds on a fence, and you hit one with a thrown rock, how many birds remain on the fence? Answer: None - they'd all start flying off when there's a rock hurtling towards them.

by rskar 14y ago

scarmig 14y ago

If I'm understanding it correctly, it's (ab)using the word "accountant" to mean "someone who can receive payments." So the question is really "Is this person someone who accepts money, or is this person someone who accepts money and is a licensed plumber?" If you interpret it literally, it uses the law of the excluded middle inappropriately and would be just as true if you replaced "accountant" with "President" or "extraterrestrial angel."

rskar 14y ago
Thanks for the reply, but I doubt that the issue surrounds (ab)use of the word "accountant". If you go to the attributed source (at http://www.reddit.com/r/math/comments/12n04d/counterintuitive_mathematical_results_accessible/c6wion3 http://www.reddit.com/r/math/comments/12n04d/counterintuitiv...), instead of water-heater/accountant/plumber it's car/teacher/mechanic. I'm more inclined to believe this is a (poor) illustration of the idea that a sub-set is likely to be smaller than the set it came from: the set of all accountants vs. the sub-set of all accountants who are also plumbers. In which case - duh! - P(accountant) > P(accountant and plumber). As long as one can indulge in the naivety of this pedagogical tale, that the narrator (the "I" who said "I went to a person") wasn't too particular on whom he hired for the job, then sure, why not?, 'twas likely the accountant that fixed it. In the real world, which has teachers and mechanics in addition to accountants and plumbers and other occupations, and where people may tend to be pickier on who they'll let near their water heater - leave alone pay someone to mess with it - would this conclusion of "accountant" still prevail necessarily?

hackinthebochs 14y ago

Your breakdown isn't quite right. Event A is "account and plumber" while Event B is "accountant" (leaving plumber status unconstrained). Thought of this way its easy to see that event B is completely contained in event A, therefore P(A) >= P(B).

rskar 14y ago
Nope, take a closer look, Event B is pictured like a crescent moon, labelled "People who are accountants and not plumbers", and described thus: "There's also the probability 'B' that the guy who fixed my water heater is only an accountant and not a plumber." Plumber status within B is clearly "Not a plumber".
- hackinthebochs 14y ago
  You're certainly right about the details you mentioned. But they are not relevant to the problem. "Accountant and not a plumber" was not in the question at all. Lets make it more precise. X = is accountant, Y = is plumber. The question is which is greater: P(X) or P(X and Y). It should be clear that the event "X and Y" is completely contained within the event "X". To put it another way, P(X) = P(X and Y) + P(X and ~Y) (we are summing over all possible unconstrained events, in this case Y and ~Y). Hence, P(X) >= P(X and Y)
  - rskar 14y ago
    Yeah, I think I picked up on that regarding P(X) >= P(X and Y) (see my reply to scarmig). From what NhanH said, I think I now see where my issue is.

NhanH 14y ago

The way it was explained in the infographic wasn't clear. But what he meant is that between P(X = accountant) and P(X = accountant and plumber), the former is bigger. The fact that P(X = plumber) > P(X = accountant) is irrelevant, as we wasn't concerned with P(X= plumber) at all. The implication of this "problem" is an interesting human biased: there appears to be cases where we judge the more detailed explanation as more probably, while statistically speaking, it isn't. If a person is described as "being concerned with discrimination and social justice", we tend to judge P(X = activist and Y) as more probable than P( X = Y). This is detailed in "Thinking, fast and slow" by Kahneman - a wonderful book if anyone haven't read it.

rskar 14y ago
I think you've shown me the way here. My issue, I suppose, is the problem with word problems in math. Generally that problem is two-fold. Does a word problem sufficiently describe the situation? And do you sufficiently understand the situation it describes? I believe this to be at the core of all the supposed "controversial math problems". BTW, thank you for suggesting Kahneman's book, it is now on my reading list. I did take notice that a sub-set is likely to be smaller than the set it came from: the set of all accountants vs. the sub-set of all accountants who are also plumbers. So therefore P(accountant) > P(accountant and plumber). Unless P(accountant) be zero, or P(plumber|accountant) be one, in which case neither is greater. Reflecting on my own human biases, I think I ended up reading it as if this were a multiple-choice question, and I had to pick the "best" choice. In other words, it isn't enough that this person is an accountant, but that this person is also a plumber. Or put another way, if I had to wager on whether anyone would pay just any old accountant to go mess with their water heater, I would find it more surprising that the particular accountant in fact hired had no demonstrable plumbing skills. I'd imagine Kahneman could teach me more on this. My favorite word problem joke goes something like this. If there are three cans on a fence, and you hit one with a thrown rock, how many cans remain on the fence? Answer: Two. Now, if there are three birds on a fence, and you hit one with a thrown rock, how many birds remain on the fence? Answer: None - they'd all start flying off when there's a rock hurtling towards them.