# Odds to clone a parent

I was asking me something weird, and I thought you might validate or not my theory.

Hypothesis: A brother and a sister have one chance over 78 billions to clone one of their parent if they procreate

Here is my method:

I have 23 chromosomes from my mother, and 23 from my father. My father has 23 * 23 = 529 combinations of chromosomes in his "gametes". If my experience is willing to work, I need to have the exact opposites of my father chromosomes than my sister ones. That means that we can't have any of my father's chromosome in common with my sister. That makes approximately 1/529 * 1/529 = 1/279 841.

Then, if I decide to procreate with my sister, our child needs to have the 46 chromosomes back from my father. So the computation stays the same and I will have 1 chance over 279 841 to make it so.

1/279841 * 1/279841 = 1/7.8*10^10


My question is: Is my theory valid? If not why? If yes, that will make a hell of a story at parties...

• For starters, you're ignoring recombination. – canadianer May 17 '17 at 21:19
• ...and mutations – Remi.b May 17 '17 at 21:43

Assumptions

You are ignoring recombination and mutations. If you were to add these things, then this probability would be extremely smaller. In apparence, it would be infinitely smaller. It makes no biological sense to make such assumption but for the sake of the argument, let's assume only segregation happen. We will also assume that the parents are completely unrelated to start with (see @BryanKrause 's comment below).

For on given pair of chromosome

First, for a given chromosome number, both parents must each transmit a different chromosome to each offspring. This happens with probability $\frac{1}{2}$ per parent and therefore $\frac{1}{4}$ for both parents. Then, if the brother transmit the maternally derived chromosome, then the sister must transmit the maternally derived chromosome as well (and vice-versa). This occurs with probability $\frac{1}{2}$. So for a given pair of chromosome the probability to clone the parent pair of chromosomes (assuming no recombination and no mutation) is $\frac{1}{4}\frac{1}{2} = \frac{1}{8}$.

For the whole genome

The probability of this to happen for all 23 pairs of chromosome is $\frac{1}{8}^{23} = 1.69 \cdot 10^{-21}$, that is one chance out of a 1.69 thousands billions billions.

As you ask for the odds, note that for such low probability the odds ratio is pretty much equal to the probability.

• Side note: your probabilities would be greatly improved if your parents, grandparents, great grandparents, and so on down the line were also each others' siblings. I wouldn't suggest it. Eventually you'd be down to just mutations, but even clones will have de novo mutations. – Bryan Krause May 17 '17 at 22:44