### Brain Teasers

# The Gardner Sisters I

Probability
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?

"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

great how u worked it out to fit wow!!!!!!!!

ya i agree with Mattroxs

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 non-blonds to pick.

The correct answer is three daughters- two blond and one non-blond. The probability of picking two blonds is the same as picking one of each ie 50%.

The correct answer is three daughters- two blond and one non-blond. The probability of picking two blonds is the same as picking one of each ie 50%.

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.

Yes you're quite right- I understand it now. Very interesting. Thanks

i did not get it wat so eva to bad 4 me!!!!!!!!!!!

yeah thats a really good one

Let there be n sisters of whom m are blond. Meeting two, the probability of both blond is then m/n * (m-1)/(n-1) = 1/2. This leads to the equation m^2 - m = (n^2-n)/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!!

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!!

I didn't use the logical answer. I just guessed!

Nice puzzle. Jimbo, I like your analysis of the solution space.

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)*((b-1)/(n-1)) = 0.5.

Then I stopped and checked the answer

(b/n)*((b-1)/(n-1)) = 0.5.

Then I stopped and checked the answer

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.

As soon as you see 3/6, you can see that b=2 and n=3.

Sorry for typo. I meant b = 3 and n = 4.

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!

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!

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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