What is an urn?

I can think of exactly two times when the word urn is used: as a container for someone’s ashes, and as a container for colored balls.  Since I’ve never physically seen an urn that has balls in it, it makes me wonder – when did that become such a standard in probability problems?  Why are the balls in an urn in the first place?

When I asked that very question, TwoPi mentioned “surmounted” as another example of an English word that seems to be used exclusively in one context:  to describe Norman Windows (a window in the shape of a rectangle surmounted by a semicircle).   “surmountable” is more common, and “insurmountable” even more so, so I suppose “surmounted” does actually appear in a related context (as in “that difficulty has been surmounted”), but it’s still relatively unusual.  I suspect that there are other words, English words as opposed to mathematical terms, that just don’t show up very often outside of the exercises in a text.

Here is a Norman Window, by the way, from Notre-Dame d’Étretat in Étretat, France.

Gene sharing math

The other day I found myself wondering what proportion of genes cousins would expect to share compared to biological siblings.  This took more time to figure out than I would have expected, in part because I knew that siblings who share two parents have 1/2 their genes in common on average, so I thought cousins sharing two grandparents might have a quarter.  They don’t, though – it’s half that.  In reasoning it out, it turned out to be easiest to think of moving up the biological family tree to a common ancestor, which led to one general formula and a few specific cases:

The Generalization:

Given two people A and B, find their closest common ancestor C.  If there are n generations from A to C and m generations from B to C, then the expected proportion of shared genes is (½)^n+m.  If there are two closest common ancestors (for example, both parents) then this number would double.

In the case of a parent and child, for example, there is 1 generation from the child to the parent (the common ancestor) and 0 from the parent to itself, so the proportion of shared genes would be (½)¹, or just ½.  Cousins would each be 2 generations from common grandparents, leading to (½)⁴, or 1/16, for cousins with one grandparent in common (sometimes called half cousins) and twice that for cousins with two grandparents in common (sometimes called full cousins).  Double cousins  — that is, people who are cousins on both sides of the family tree (for example, cousins whose mothers are sisters and whose fathers are brothers) — would still have grandparents as the closest common ancestor, but now it would be up to four common grandparents instead of just one or two: the expected proportion of shared genes between cousins with four common grandparents would be 4·(½)⁴, or just ¼.  Likewise, an aunt and nephew with two parents/grandparents in common would be 1 and 2 generations respectively from this pair of common ancestors, so the expected proportion of shared genes would be 2·(½)³, also ¼.

Special Case 1:  great-great-…-great grandparents
In this case the older relative is the common ancestor, so if “g” is the number of “great”s then the proportion of shared genes is (½)^g+2.  The additional 2 in the exponent is because the number of “great”s counts the generations after grandparents, who are already 2 generations away from their grandchildren.  This is the only case where the proportion is exact:  in all the others, it’s only an expected proportion because siblings could have anywhere from no overlap of genes to complete overlap of genes from each common parent.

Special Case 2:  great-great-…-great aunts and uncles
In this case the older relative’s parent(s) are the common ancestor.  With a great-uncle and great-niece, for example, the great-uncle’s parent(s) are the great-grandparent(s) of the great-nephew.  This means that there is 1 generation from the great-uncle to his parent(s), but 3 from the great-niece to that common ancestor, with each additional “great” adding another generation.  If “g” is the number of “great”s, then the expected proportion of shared genes would be (½)^g+3 if there is one parent in common, and (½)^g+2 if there are two.  (I personally find it interesting that you can expect to share the same proportion of genes with a sibling who shares both parents as you do with either of the individual parents, the same proportion with an aunt or uncle who shares both grandparents as you do with either of the individual grandparents, and the same proportion with a great-great-…-great aunt/uncle who shares both great-great-…-great grandparents as you do with either of those great-great-…-great grandparents themselves.)

One clarification:  great-aunt is the term I grew up with, but in looking around I just discovered that “grand-aunt” may be the technically correct term, since that person is in the same generation as a grandparent; likewise, the sister of a great-grandparent would be a great-grand-aunt.  This appeals to me aesthetically.   If you were to use these terms, then you’d have one fewer “great” in describing the relationship, and you’d need to add 1 to the exponent in the formulas above.

Special Case 3:  second cousins once removed (and the like)
Cousins share at least one grandparent, second cousins share at least one great-grandparent, and x^th cousins share at least one great^(x-1) grandparents.  This means that x^th cousins are each (x+1) generation removed from the common ancestor(s), and would expect to share (½)^2x+2 of their genes if there is one common relative and (½)^2x+1 if there are two. Each removal  refers to one of the people being one more generation removed from any common ancestors, and so increases the power of ½ by 1.  This means that x^th cousins who are y-times removed would expect to share (½)^2x+y+2 of their genes if there is one common relative and (½)^2x+y+1 if there are two. Second cousins once removed would share either (½)⁷ or (½)⁶ of their genes, while first cousins twice removed would share (½)⁶ or (½)⁵.

For those who like the visual, there is a handy little chart below, which appears to be in the public domain on Wikipedia. It does make some assumptions, however – namely, that siblings, cousins, aunts and nieces, etc. have exactly two closest relatives in common (both parents, two grandparents, etc.).

Soccer Math

The World Cup is happening! It’s inspiring to watch excellent soccer players…inspiring us to write about some excellent math. We’ll venture out on the field with a few soccer and math tidbits.

• The soccer ball that I think of as typical – in other words, the one that I remember from Days of Yore – is an Archimedian solid, made from 20 regular hexagons and 12 regular pentagons.  Specifically, it’s a truncated icosahedron because it can be built from lopping the corners off of a regular icosahedron.  It’s also a buckminsterfullerene, although that’s only the formal name:  friends can call it a buckyball.  The buckyball was Red Hot News in 1985, because it was a new way of putting Carbon atoms together.  Scientists Harold W. Kroto, Robert F. Curl, and Richard E. Smalley* named it after architect  R. Buckminster Fuller*, whose geodesic domes had inspired them to try and create such a carbon cluster.   But this soccerball-shaped soccerball doesn’t limit itself to Ancient Greeks and Modern Scientists, oh no.  It also dabbles in the arts, as shown in this photo below from Labor Park in Dalien, China.
• Soccer balls aren’t the only thing math-related in soccer: there’s also the number of people on a team.  Each team has 23 players, which means that on any team there is a 50% chance that two people on that team share a birthday.  With 32 teams playing in the world cup, you’d expect about half of them to have birthday-sharing teammates, and in fact, as the BBC pointed out earlier this week,  exactly 16 of the 32 teams do.    For example, tomorrow (June 20) six people have birthdays, including two (Asmir Begovic and Sead Kolasinac) on the team from Bosnia and Herzegovina.  Now oddly enough, even though you’d expect half the teams to have teammates sharing a birthday, the fact that it’s exactly half is actually rather strange:  with a 50% chance of two teammates getting to share cake, the probability that exactly 16 of the 32 teams satisfy that is only 14% – it’s just that at that point it’s equally likely to be more or fewer days.  Ironically, it’s rather unexpected to actually hit the expected value.
• One final math fact about the World Cup: one of the referees for yesterday’s match between Chile and Spain is actually a former high school math teacher!    Not all that former, either:  Mark Geiger taught in New Jersey alongside his brother, winning the Presidential Award for Excellence in Math and Science Teaching, but eighteen months ago he left teaching in order to referee full-time, hoping for a shot at the World Cup.  Not a bad gig, and he always has those math skills to fall back on if he finds he misses teaching.

*Whenever I type “[Occupation] [Person’s Name]” I get the urge to add “renowned” and then go read Da Vinci Code again.

The photo of the sculpture is by Uwe Aranas, Creative Commons License.  And if you didn’t follow the link to the BBC article, “The Birthday Paradox ath the World Cup” by James Fletcher, it’s worth a read – it has a lot more detail about the birthday paradox and sports.