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In a certain interview this question was asked to one of my friends by the interviewer

what is the probability of this running fan (pointing at a ceiling fan) to fall down?

Actually to find the answer I feel that sample space is two

a. Either the fan will stay at its position. b. Or The fan will fall down. So what I feel, probability is 50%

Please let me know whether this is true or false, guide me for the thinking. I'm also going to sit for similar kind of interview. Please help

Sourav
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    Either there will be a mouse-sized dinosaur dancing on my bed when I get home tonight, or there will not be. Hence 50-50? –  Jun 18 '14 at 04:02
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    So, it's just as likely as all the bolts shear and the fan collapses as not? That would mean every other building should collapse daily, no? Just because the solution set contains two possibilities doesn't mean that those two are equally likely to happen. – Avraham Jun 18 '14 at 04:08
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    When? In the next half-hour? – André Nicolas Jun 18 '14 at 04:09
  • @AndréNicolas: +1. The question is vacuous without specifying a time interval. All I can tell you at an instant is the density of the probability, which isn't what is asked. – Eric Towers Jun 18 '14 at 04:19
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    Very closely related to this question: Why is not the answer to all probability questions 1/2. – ShreevatsaR Jun 18 '14 at 04:44
  • This might refer to Laplace's argument. Then the fan falls down tomorrow with a probability roughly equal to the inverse of the number of days since it was installed. – Did Jun 18 '14 at 05:39
  • Obviously, this is not a question on testing one’s mathematical knowledge because they did not provide any related information to that. Its purposes are (1) testing one’s reaction; (2) testing one’s “confidence level”. For the latter, I would say “the probability is 0, … if it was installed 100% by me.” – Mick Jun 18 '14 at 05:59

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A distribution need not be uniform. The sample space for the sum of two dice is 11: $\{2,3,4,5,6,7,8,9,10,11,12\}$, but $2$ and $12$ can only happen a single way, whereas there are six ways a $7$ can be rolled.

Of course, most likely this question has no right answer, because it is an "interview 2.0" question.

However, if you actually were being hired as a reliability engineer for ceiling fan mounts, the appropriate answer is that you would have to perform a maximum likelihood analysis using right-censored data: since you don't know the time-to-failure of fans that have not yet failed, it is impossible to completely describe the distribution. Instead, we must make some assumptions on the tail distribution of the right tail of failure times and proceed from there.

Emily
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You have several answers available to you:

  • I have never seen this model of fan mount fail. My prior expectation of failure is near zero and only deviates from zero because I am willing to entertain the notion that this mount or its installation is somehow at variance with my prior experience. (This is a Bayesian argument.)
  • (Your 50% is workable, but not the way you used it.) The Copernican Argument is that there is nothing special about the time you have observed the fan. Since it has not fallen during the past (however long you have been in the room, say 20 minutes, but use whatever your estimate of the real amount of time is) I predict with 50% confidence from the data so far that it will not fall in the next 20 minutes. (The idea is that your observation has no special place in the history of the object (just as the Earth is not at a special place: the center of the universe) so you are equally likely to be observing it anywhere along that duration.) If I knew that it had been mounted continuously for time prior to my presence in the room, I would adjust my estimate accordingly.
  • 100%. Nothing lasts forever. (This, depending on your outlook on like is either the optimistic, pessimistic, or realist position.)
  • There are more ways to go, but this sort of question is just to see how much of a deer in the headlights response you produce. Just don't let your brain seize up when something unexpected happens, as that might indicate a problem.
Eric Towers
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