There are two parcels, one marked CALCUTTA and one marked TATANAGAR which get lost in transit. One parcel is found and has the label partly torn (the label can be torn in any manner). It only has the letters 'TA' (adjacent letters). What is the probability that the recovered parcel was marked CALCUTTA?

(In reply to re(2): Posterior thinking (spoiler?) by Vishal Gupta)

I have now had further thoughts about this.

We have to assume that out of all the ways the labels could be torn, those that leave 2 and exactly 2 letters can be assigned exactly the same probability of occurrence (of which more below). Then what we are doing is comparing the frequency of (TA given TATANAGAR = 1/4) with the frequency of (TA given CALCUTTA= 1/7). That tells us that it is 4/7 times as likely that the parcel came from Calcutta as it is that it came from Tatanagar, rather than half as likely, as one might intuitively suppose. Now there are 11 chances, of which 4 are Calcutta, and 7 Tatanagar, resulting in the probabilities 4/11, 7/11.

I see a big problem with the first step, though. It's quite likely that the label could have been affixed at the ends, so that the initial TA or the last TA could have been torn off, but the second TA would require (at least) 2 (bizarrely accurate) tears. Is this really exactly as likely as one tear?  If we assume that it's even just half as likely (which I doubt), then there are 37 chances, of which 16 are Calcutta, and 21 Tatanagar.

So If I absulutely had to bet on this, I'd guess around 43.25% Calcutta, 56.75% Tatanagar as the maximum variance between the probabilities; but this is not that much different from the 50/50 chance that one could get without all this effort!

I might add that this last issue could be avoided by using somewhere such as Tahta, Egypt, or Tatta, Pakistan instead of Tatanagar - but I suppose the difference in the stamp would then be a giveaway!

 

Edited on October 29, 2010, 8:17 am

Posted by broll on 2010-10-29 08:04:42