md65536 Posted June 22, 2012 Posted June 22, 2012 A farmer (on an island of course) has 100 cows in a barn. Each is brown or blue, and is equally likely to be one or the other. A cow escapes from the barn and it is brown. What is the expected proportion of brown cows to total cows? There is a trick to the wording of this one...
Joatmon Posted June 22, 2012 Posted June 22, 2012 (edited) Reveal hidden contents I will ask the obvious question - Just in the barn or on the island plus barn? Edited June 22, 2012 by Joatmon
md65536 Posted June 22, 2012 Author Posted June 22, 2012 On 6/22/2012 at 6:38 PM, Joatmon said: Reveal hidden contents I will ask the obvious question - Just in the barn or on the island plus barn? I meant the proportion of the 100 cows. The same applies to the other 2 puzzles! I didn't mean for that to be part of the trick.
Greg H. Posted June 22, 2012 Posted June 22, 2012 Reveal hidden contents Why does it matter if the barn is brown?
md65536 Posted June 22, 2012 Author Posted June 22, 2012 On 6/22/2012 at 7:03 PM, Greg H. said: Reveal hidden contents Why does it matter if the barn is brown? It doesn't. Actually the barn is hot pink and the escaped cow is brown.
Joatmon Posted June 22, 2012 Posted June 22, 2012 Reveal hidden contents Does the farmer own any cows that aren't in the barn before the one escapes?
md65536 Posted June 22, 2012 Author Posted June 22, 2012 On 6/22/2012 at 10:14 PM, Joatmon said: Reveal hidden contents Does the farmer own any cows that aren't in the barn before the one escapes? No, the "trick" involves stats/probability stuff, and not the physical details of the imaginary world. To be honest, I changed my mind about the answer I had in mind after I posted the question. So I don't even know if this is a valid puzzle!, or what the solution is...
md65536 Posted June 23, 2012 Author Posted June 23, 2012 (edited) What if I changed the question to... Reveal hidden contents There are 100 cows, each either blue or brown. One cow (of the 100) in particular is brown. Each of the 100 cows is equally likely to be blue or brown. What is the expected proportion of brown cows to total cows? Edited June 23, 2012 by md65536
Willa Posted June 25, 2012 Posted June 25, 2012 Reveal hidden contents So, we are to find the expected value of (brown cows)/100, given that Cow #1 is brown. Since the color of each cow is independent of that of the other cows, this basically means the first cow is brown with 100% probability while each of the others is brown with 50% probability. Therefore the expected number of brown cows should be 1 + .5*99 = 50.5. So the answer is .505.
md65536 Posted June 25, 2012 Author Posted June 25, 2012 On 6/25/2012 at 8:39 PM, Willa said: Reveal hidden contents the first cow is brown with 100% probability while each of the others is brown with 50% probability. Therefore the expected number of brown cows should be 1 + .5*99 = 50.5. So the answer is .505. That's certainly the most sensible interpretation of the puzzle... But: Reveal hidden contents The puzzle doesn't say that the probability of any of the 99 others being brown is 50%, it says that the probability of any (of the full 100) being brown is 50%. Does the only acceptable interpretation of a statement like "Each is equally likely to be one or the other" include the caveat that "this is true only in the absence of additional information"? Or is it valid to interpret the puzzle as impossible or nonsense, eg. with a statement like "The brown cow is equally likely to be brown or blue."? I don't know the answer to that!!!
Willa Posted June 25, 2012 Posted June 25, 2012 Reveal hidden contents Aha. So that's the "trick" in the wording. I think the most natural way to interpret the problem is sequentially: First, while all the cows are in the barn out of sight, each of the 100 has an equal chance of being brown or blue. Then, when you observe the brown cow escaping, the probability of that particular cow being brown changes from 50% to 100%. Assumptions here: Bayseian (belief-based) interpretation of probability The barn is opaque/you can't determine the color of cows inside it Under this interpretation, my answer of .505 holds.
md65536 Posted June 25, 2012 Author Posted June 25, 2012 On 6/25/2012 at 10:31 PM, Willa said: Reveal hidden contents Assumptions here: Bayseian (belief-based) interpretation of probability Yes I think that must be the key. The statement about a cow's 50% likelihood of being brown isn't a statement of absolute permanent fact (every actual cow has an actual color, it's just not known) but a statement of belief or uncertainty that must be open to additional information. I guess any interpretation other than yours would be a "too literal" misinterpretation. Thanks, this has been more of a lesson for me than a real puzzle.
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