Math Problem: What percent of us are related?
March 27th, 2015 at 8:52:54 AM
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I was recently thinking about family trees and ancestry trees and came up with this generalized question:
Suppose an ancestry tree starts at the very top with n unrelated people. Each of these n people (and future generations) get involved in, on average, r relationships (r might be 0.75 if many people don't have any relationships or 1.2 if many people have multiple relationships). Assume people only get involved in relationships with other people in their own generation level. Each relationship produces, on average, c children.
If the ancestry tree has a total of g generations (the top level is generation 0), answer the following questions:
1 (easy). How many people will there be at generation level g?
2 (medium). If two different people are chosen at random in generation level g, what is the percent chance that they are related (share a common ancestor). Assume for this part that there is no "incest" (i.e. no two people who are related get involved in a relationship)
3 (hard). Same as problem 2, but make no assumptions about "incest".
4 (super hard?). Bonus: Same as problem 2, but "related" as far as "incest" is concerned means that two people share an ancestor less than i generations ago. Can this version of the problem be solved with the information given?
Note: I have a solution to parts 1 and 2, but have yet to determine if parts 3 and 4 can be solved or not. There might be a simplification that I'm not aware of.
Hint 1:
Hint 2:
Suppose an ancestry tree starts at the very top with n unrelated people. Each of these n people (and future generations) get involved in, on average, r relationships (r might be 0.75 if many people don't have any relationships or 1.2 if many people have multiple relationships). Assume people only get involved in relationships with other people in their own generation level. Each relationship produces, on average, c children.
If the ancestry tree has a total of g generations (the top level is generation 0), answer the following questions:
1 (easy). How many people will there be at generation level g?
2 (medium). If two different people are chosen at random in generation level g, what is the percent chance that they are related (share a common ancestor). Assume for this part that there is no "incest" (i.e. no two people who are related get involved in a relationship)
3 (hard). Same as problem 2, but make no assumptions about "incest".
4 (super hard?). Bonus: Same as problem 2, but "related" as far as "incest" is concerned means that two people share an ancestor less than i generations ago. Can this version of the problem be solved with the information given?
Note: I have a solution to parts 1 and 2, but have yet to determine if parts 3 and 4 can be solved or not. There might be a simplification that I'm not aware of.
Hint 1:
Note that each relationship involves two people, so if you say n*r relationships occur at generation 0, you're double counting every relationship.
Hint 2:
#2 can be solved easiest using a bottom up approach (i.e. starting with the two people picked and finding their ancestors back to generation 0). #3 must be solved (I think) top down (starting at generation 0 finding their descendants down to generation g).
March 27th, 2015 at 8:58:46 AM
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If needed, assume that the amount of relationships and children follow the normal distribution centered at r and c respectively.
March 27th, 2015 at 6:57:49 PM
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These are good questions but would be difficult to answer. If I were to tackle this I would use a computer simulation.
On a practical level, I think the day is not far off when any two people can submit a DNA sample with a Q-tip scraped against the inside of the cheek and they can be compared for a percentage of common DNA. For example, two siblings would have 50%, two cousins 12.5%.
On a practical level, I think the day is not far off when any two people can submit a DNA sample with a Q-tip scraped against the inside of the cheek and they can be compared for a percentage of common DNA. For example, two siblings would have 50%, two cousins 12.5%.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
March 27th, 2015 at 7:57:37 PM
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Yes, parts 3 and 4 seem to be rather difficult to solve directly with math.
I think parts 1 and 2 are similar in difficulty to other questions on this forum, though.
I think parts 1 and 2 are similar in difficulty to other questions on this forum, though.
March 28th, 2015 at 10:18:29 AM
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Quote: WizardOn a practical level, I think the day is not far off when any two people can submit a DNA sample with a Q-tip scraped against the inside of the cheek and they can be compared for a percentage of common DNA. For example, two siblings would have 50%, two cousins 12.5%.
This video just showed up in my YouTube feed. Freaky how much my computer knows about me.
