# Flipping coins

This is not a homework question. It is actually kind of a biology question, but I'm dealing with the pure mathematics here. We came up with this reading "The selfish gene" by Richard Dawkins. He writes something like: "Elizabeth II is a direct decsendant William the conqueror. However it is likely that they share not a single gene." This struck us as odd.

Being the only person in the room who knew advanced mathematics I took on the problem. Now I'd like to check if my solution is correct, because I feel I've got some flaws.

What my biologist friends told me is that one inherits 25000 genes from ones father. Also we counted that there are 30 generations between William and Elizabeth.

I think this would be analogous to flipping 25000 coins. All that end up heads one discards, and the ones that end up tail gets flipped again. What is the probability that after 29 throws we don't have a single coin that came up tail every time?

Doing some calculations I came up with a formula

$$25000\cdot0,5-\sum_{k=1}^{25000}(25000-k)0,5^{k+1}$$

This should at least give the right answer for the first generation. Can I then just raise it to the power of 28 or should i do something different. Also, one problem is that when i tried to program this into matlab the first generation had a probability so close to one, it was one. Then rasing it to the power of anything won't give an accurate answer. This would say Dawkins was way of.

Any ideas?

## Answers and Replies

Actually, thinking of this in another way says that because of the law of large numbers about half of the coins will end up heads. Therefore we can calculate this as $$25000\cdot0,5^{29}=4,66\cdot10^{-5}$$. Although if we think of how many ancestors one should have so nobody reproduces with someone that's even remotely related we end up with 536870912 persons in the 29th generation.

CRGreathouse
Science Advisor
Homework Helper
Your understanding of math is fine. Your understanding of biology, on the other hand... :D
What my biologist friends told me is that one inherits 25000 genes from ones father. Also we counted that there are 30 generations between William and Elizabeth.

Genes are inherited via chromosomes, of which a parent passes on only 23.

William shared 46 chromosomes with William. William's children shared 23. Their children shared between 0 (probability 2^-23) and 23 (probability 2^-23) with William.

CRGreathouse
Science Advisor
Homework Helper
The probability I get that William passed no genes directly to Elizabeth (though she might have had some of his from other ancestors!) assuming no interbreeding or polysomy is 99.9999957%.

Code:
generation(v)=v=vector(23,i,v[i]/2^i);vector(23,i,sum(j=i,23,binomial(j,i)*v[j]))
gens(n)=my(v=vector(23,i,i==23));for(i=1,n,v=generation(v));v
G=gens(29);
1.-sum(i=1,23,G[i])

OK. Thank you. I figured I got something incorrectly. Does this mean that if William passed genes to Elizabeth he passed a complete chromosome? And that happened with 99.9999957% possibility? Just curious.

CRGreathouse
Science Advisor
Homework Helper
OK. Thank you. I figured I got something incorrectly. Does this mean that if William passed genes to Elizabeth he passed a complete chromosome? And that happened with 99.9999957% possibility? Just curious.

If William passed on any genes, he passed along at least one whole chromosome (so that Elizabeth would be "over 2% William"). The probability that this would happen is 0.0000043%; the probability that he passed on zero chromosomes is 99.9999957%.