Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- Blue Eyes
Alex Y
Another Gardner puzzle.
If you meet two of the Jones sisters (chosen at random from a single nuclear family), there is a 50-50 chance that they are both blue-eyed. How many sisters are there likely to be in this family?
For extra credit, in the unlikely event that the answer to the above is not the number of sisters, then how many are there?
Re: Friday Puzzler -- Blue Eyes
John Veerkamp
There were 6 of us siblings. 2 had blue eyes, the rest variations of brown. Based on this very small sample(ain't statistics wonderful) there were eight sisters.
Re: Friday Puzzler -- Blue Eyes
Larry Barrett
I get a different answer than John, at least to the first part of the puzzle.
I think there are 4 sisters, and three have blue eyes. Number the sisters 1, 2, 3, 4; 1, 2, and 3 are the blue-eyed ones. Then the combinations of two selected at random that result in both having blue eyes are 1-2, 1-3, 2-3. The total set of combinations of pairs is 1-2, 1-3- 1-4, 2-3, 2-4, 3-4. The odds of selecting a pair tht are both blue-eyed = 3/6 = .5.
I don't yet understand the second part of the problem.
Correct
Alex Y
While the answer you gave is by far the most likely, there are other possibilities for the count of Jones sisters. I was asking about the second most likely (although very unlikely) possibility.I don't yet understand the second part of the problem.
You've enumerated the possibilities for the sisters I met in the family with four girls, three of whom have blue eyes. In general, if there are n sisters and k of them have blue eyes, then there are n*(n-1)/2 ways of choosing n sisters and there are k*(k-1)/2 ways of choosing two blue-eyed sisters, so the problem reduces to finding n and k such that k*(k-1) = .5*n*(n-1).
By far the most likely is 4 sisters, 3 of whom are blue-eyed. The next most likely is 21 sisters, 15 of whom have blue eyes, then we go to 120/85 sisters, then 697/493 sisters, and presumably infinitely many possibilities above that.