The Gardner Sisters I
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.
Gretchen and Henry were discussing their new neighbors, the Gardners. Gretchen mentioned that she met two of the daughters, and they each had blond hair.
"I have met all of the sisters," replied Henry, "and the probability that both of the girls you met would have had blond hair, assuming you were equally likely to meet any of the sisters, is exactly 50%. Do you know how many children there are?"
After thinking for a minute, Gretchen asks if the family is abnormally large. When Henry replies that it is not, Gretchen tells him how many girls are in the family. What number did she say?
Answer
Gretchen said that there were 4 girls in the family, three of whom were blond.
This would make the probability that she saw two blonds (3/4) * (2/3), which equals 1/2.
Other numbers would work, but the next pair would lead to a rather large family.
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Comments
Mattroxs
Aug 28, 2006
 great how u worked it out to fit wow!!!!!!!! 
BLACKlover
Sep 01, 2006
 ya i agree with Mattroxs 
harryharry
Dec 29, 2006
 This is wrong you are expressing a certainy as a probability of less than 1. If there are four girls, three of whom are blond and you meet two of them, then the probability of one of them being blond is 100%. There simply aren't enough nonblonds to pick.
The correct answer is three daughters two blond and one nonblond. The probability of picking two blonds is the same as picking one of each ie 50%. 
tsimkin
Dec 29, 2006
 harryharry  The problem states that the probability that they are BOTH blond is 50%. I agree that the probability that AT LEAST ONE is blond is 100%, but that is irrelevant. 
harryharry
Dec 30, 2006
 Yes you're quite right I understand it now. Very interesting. Thanks 
shopaholic
Mar 04, 2007
 i did not get it wat so eva to bad 4 me!!!!!!!!!!! 
TiffyTheGreat1
Apr 14, 2007
 yeah thats a really good one

Jimbo
Apr 29, 2007
 Let there be n sisters of whom m are blond. Meeting two, the probability of both blond is then m/n * (m1)/(n1) = 1/2. This leads to the equation m^2  m = (n^2n)/2. Now for any given n, can we find integer solutions for m. From the puzzle, n is at least 3.
n=3: m^2  m  3 = 0. No integer solutions.
n=4: m^2  m  6 = 0. Yes m = 3.
n=5: m^2  m  10 = 0. No integer solutions.
n=6: m^2  m  15 = 0. No integer solutions.
etc. We are basically looking for a triangular number (3,6,10,15 etc) which has factors that differ by one. We have found 2*3=6 so 3 is a solution.
The triangular numbers continue..(21, 28, 36, 45, 55, 66, 78, 91, 105, 120, ...) At this point we can probably stop looking because we are up to the 15th triangular number without finding another solution and the family was not inordinately large.
Loved it!! 
PRAragorn
Jul 16, 2007
 I didn't use the logical answer. I just guessed! 
javaguru
Dec 09, 2008
 Nice puzzle. Jimbo, I like your analysis of the solution space. 
Stack1607
Jan 06, 2010
 I reached the conclusion that if they had a total number of daughters equal to n, from whom b are blond. Then the following equation holds true:
(b/n)*((b1)/(n1)) = 0.5.
Then I stopped and checked the answer 
opqpop
Sep 16, 2010
 Let there be n girls and b of them blonde. The total possible ways to meet 2 girls is n choose 2. The total possible ways to meet 2 blonde girls is b choose 2. List a few k choose 2 integers for k = 2, 3, 4, ... and you get 1, 3, 6, ...
As soon as you see 3/6, you can see that b=2 and n=3. 
opqpop
Sep 16, 2010
 Sorry for typo. I meant b = 3 and n = 4. 
bigbanggoo
Jun 28, 2011
 This is WRONG! Gretchen told him how many girls ARE IN THE FAMILY not how many sisters there are. Gretchen did not answer the question.
Also even if there really are 4 girls in the children then that means that there are at least 5 girls in the family including the MOTHER! 4 girls, half of which she saw( the two blonds) and then the mother! 
tsimkin
Jun 28, 2011
 Bigbanggoo  thank you for your constructive help on this problem. I should be clear  I was using the generally accepted form of "girl" meaning "female child, from birth through adolescence". It is true that with the wider construct of "girl" meaning "any female", you should include the mother. It is also true that this family may have had pets, who themselves may have been female, and therefore should also be included in the number of "girls" Gretchen mentions at the end. I am appropriately chastened by your insight. Thank you for taking the time to add value to this question and this site. 
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