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This is a very interesting word problem that I came across in an old textbook of mine. So I know it's got something to do with probability, but other than that, the textbook gave no hints really and I'm really not sure about how to approach it. Probability has always given me headaches but in fact, I've been racking my head over it so long that I've got a huge headache. Any guidance hints or help would be truly greatly appreciated.

"Last week, the heat in my apartment crapped out because my water heater broke.

I went to a person, showed him the water heater and asked him to fix it."

Ignoring the practicality side of this problem and instead focusing on the math probability side of it,

Is this person more likely:

  • An accountant

OR

  • An accountant and a plumber?
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anonymous
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    @SimonS : I think you are mistaken. The answer posted by "DonkeyKong" below makes it clear. This is actually a somewhat standard exercise in probability. ${}\qquad{}$ – Michael Hardy Jun 20 '15 at 03:39
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    Haha the same argument is given in Daniel Kahneman's "Thinking Fast and Slow", although its a psychology / behavioral economics book and not really a math one. I highly recommend it nonetheless! – Faraz Masroor Jun 20 '15 at 04:10
  • @FarazMasroor's comment lead me to this Wikipedia article: Conjunction fallacy – Martin Sleziak Jun 20 '15 at 05:45
  • @MartinSleziak Thank you for that. It has some of the necessary information. – anonymous Jun 20 '15 at 05:46
  • Two similar older posts: The water heater problem ( mathematician or plumber)?? and Probability Question: Would A always have a greater chance of A?B? – Martin Sleziak Jun 20 '15 at 11:05
  • @SimonS I completely agree that the question would be less intentionally misleading/requiring of less assumptions if it weren't "plumbers". "Plumbers" were probably just chosen to try and illustrate the fallacy. Also, *Donkey Kong's :) – miradulo Jun 20 '15 at 16:34
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    @SimonS : The probability that an accountant can fix a water heater has nothing to do with this. If you digest "Donkey Kong"'s answer, you'll see where the actual math problem is in the posted question. – Michael Hardy Jun 20 '15 at 17:18
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    The question is internally inconsistent: on the one hand, it asks us to ignore the practicality side of the question, and OTOH, it goes to great pains to specify that the work to be done is a PRACTICAL problem that requires practical plumber skills, and further that one of the people has such practical plumber skills. /// Make up your mind. If one wishes to pose the problem without the practical side, then by all means do so ... the question then reduces to: I met a random person. Is his profession more likely A ... or A AND B? That is a completely different question to that posed here. – wolfies Jun 20 '15 at 18:09
  • @wolfies Indeed. See my answer below. – Ethan Bolker Jun 20 '15 at 18:37
  • I give up. Deleting all comments. – Simon S Jun 20 '15 at 18:43
  • I've noticed that you have 11 questions in your first 7 days on the site. I wanted to make sure that you are aware of the quotas 50 questions/30 days and 6 questions/24 hours, so that you can plan posting your questions accordingly. (If you try to post more questions, StackExchange software will not allow you to do so.) For more details see meta. – Martin Sleziak Jun 20 '15 at 19:54
  • @MartinSleziak Oh ok I will ask less questions then. – anonymous Jun 20 '15 at 21:01
  • I have no problem with you asking questions. (Ask as many as you like.) I just wanted to make you aware of the limitations imposed by the site. (So that, for example, if a user asks 5 questions a day, they might be surprised that after 10 days they will not be allowed to post a new question. But with your current rate you are probably still safely within the bounds.) – Martin Sleziak Jun 20 '15 at 21:09
  • @MartinSleziak Thank you. – anonymous Jun 20 '15 at 21:18

2 Answers2

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Simple set theory, being an accountant and a plumber is a subset of being an accountant, so the probability of being an accountant is greater.

To put it in other words, every person who is an accountant and a plumber is indeed an accountant! But not all accountants are also plumbers.

anonymous
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miradulo
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I've seen this kind of question and answer before. Let me venture a contrary (Bayesian) opinion.

The story gives you important prior information - you already know he's a plumber (with very high probability - he might just be a talented amateur). With that prior information, the universe consists only of plumbers. In that universe the probability of (accountant and plumber) is essentially the same as the probability of (accountant). It's probably pretty low.

Framing the question as an instance of the conjunction fallacy asks you to ignore the fact that you know he's a plumber.

The point of my answer is to try to save the OP from his comment:

God I'm so dumb...I always confuse myself over the simple questions. Now it makes sense. – anonymous 19 hours ago

The letter of the probability laws say the answer is as @DonkeyKong correctly notes. I'm looking for a formulation that is mathematically defensible and matches common sense. In answer to @Rahul 's comment, consider a weighted average. Suppose the probability that the repairman is a talented amateur is 0.01. That's the case in which the probability of (A and P) is less than the probability of P. The other 99% of the time the probability of P is 1 and the probability of (A and P) is the probability of A in the universe of plumbers, which is much less than 1.

Ethan Bolker
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  • Interesting. I don't really see how this is a contrary opinion though, your assumption that everyone is a plumber just leaves you with plumbers representing the whole set, and a subsequent equality for any intersection with plumbers. Same concept (I think). – miradulo Jun 20 '15 at 14:01
  • @DonkeyKong It's a contrary opinion because it arrives at a different conclusion. I agree that the probability computations work the same way once you've specified the whole set. – Ethan Bolker Jun 20 '15 at 14:20
  • That's fair enough! – miradulo Jun 20 '15 at 14:22
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    But as you admit, you don't know that he's a plumber -- you only know it with high probability. There is a small chance that he might just be a talented amateur (and also an accountant). So the probability of being accountant is still greater than that of being an accountant and a plumber. –  Jun 20 '15 at 20:08
  • @Rahul See my edit. It takes a little temerity to argue with someone who comments regularly on Andrew Gelman's blog. – Ethan Bolker Jun 21 '15 at 00:44
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    Don't worry, "Rahul" is a very common name in India so you should not assume that all those comments on Dr. Gelman's blog are mine! –  Jun 21 '15 at 02:09