Puppy Perplexity

Here is yet a different kind of puzzle. It will teach you (if you figure it out) how to solve another large class of logic puzzles. Enjoy. It isn't as easy as you might think!

Your Yellow Lab Retriever is having puppies!! You (Bill) watch as she quickly delivers two males. You run upstairs to grab your camera and the doorbell rings - two of your friends (Joe and Bob) have come over to visit. You mention that your dog has just given birth. After a little while you all go downstairs to see how the they are doing. On the way down, you mention to Joe (Bob doesn't hear) that there are two male puppies. When you return, a third has been born... a chocolate!! You now have one of each color. The yellow and black lab puppies (your friends don't know in which order they were born) tumble over each other and Joe notices that those two are boys. You challenge each other: What is the probability that all three puppies are male? Bob overhears the challenge, but he doesn't know the gender of any of the puppies.

Bill, Joe, and Bob all happen to be taking "Statistics 101" together at the local community collage. Over a beer they jot down their answers. They are all pretty bright students (assume they get it right) and competitive (they don't help each other or compare notes). What did they come up with?

If you give up, here is the solution:
http://blogs.sun.com/roller/resources/dcb/SOLUTION_Puppy_Perplexity.html

Comments:

Doesn't Joe have two pieces of information? He has observed two puppies which were both boys and he knows that Bill observed two puppies which were both boys. This bifurcates the possible states into states where Joe and Bill observed the same puppies (probability 1/3) and states where they observed different puppies (2/3). Joe deduces that the probability is 1/2 if they observed the same puppies and 1 if they observed different puppies, so his answer should be 1/3\*1/2 + 2/3\*1 = 5/6. Consider a situation where 1000 people arrive sequentially. Each of them looks at two puppies and sees two boys, and each of them is told that all of the previous arrivals also looked at two and saw two boys. The last person can be pretty confident that if nobody saw a girl there just isn't one. You can also take it to the limit with 1000 puppies. The chances that Joe looks at exactly the first 999 puppies born is pretty low, so he should deduce that he probably saw a boy that was not seen by Bill and hence all of the puppies are boys.

Posted by Nathan Bronson on March 03, 2005 at 07:22 AM EST #

I think Nathan is correct. We have to also assume that Bill doesn't see the two boy puppies playing, or else he would know 100% what the situation is, assuming he remembered the colors of the first two puppies.

Posted by Kevin on March 04, 2005 at 08:54 AM EST #

Bill did see the two being born. Yellow and Black. And saw that \*those\* were the two playing. Therefore, still has no clue to the gender of the chocolate, and there is a 50/50 chance of that one (and therefore, all three) being male. Bill's odds are 50%. Bob's odds are 12.5%. However, I was wrong about Joe. His odds are also 50%. I'll update my solution.

Posted by Dave Brillhart on March 04, 2005 at 09:03 AM EST #

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