# Per stirpes and random walks

If an inheritance is to be divided per stirpes, each descendant gets an equal share. If a descendant has died but has living descendants, his or her share is distributed by applying the rule recursively.

## Example

For example, suppose a man had two children, Alice and Bob, and stipulates in his will that his estate is to be divided per stirpes. If Alice and Bob are still alive when he dies, his estate is split evenly. Suppose, however, that Alice is still alive but Bob has died, and that Bob has three living children, Carol, David, and Erin. In that case Alice would inherit 1/2 of the estate, and each of Bob’s children would inherit 1/6, i.e. 1/3 of Bob’s 1/2.

## State law

In some states, such as Texas, per stirpes is the default when someone dies without a will. Who knows what they do in Nevada? Maybe the descendants play poker for the inheritance. I don’t know. I’m not a lawyer, certainly not a Nevada lawyer.

## Random walk

Here’s a random process whose expected value gives the same result as per stirpes.

Convert the inheritance to a jar of pennies, possibly a very large jar. Repeat the following steps until all the pennies are gone.

1. Take a penny out of the jar and perform a random walk on the family tree representing the descendants of the departed.
2. When you come to a branch in the tree, choose a branch at random with each branch having equal probability.
3. When you encounter a living person, give them the penny.

This assumes that you first prune the descendant tree of any lines that have died out. That is, we assume every terminal node of the tree represents a living person.

Why is it necessary to trim the tree? If you ended up at a dead end, couldn’t you just put the penny back and start over? No. Suppose in the example above that Alice and Carol are the only surviving descendants. Then per stirpes says they should receive equal amounts, since Carol inherits all of her father’s share. But if we did the random walk without removing David and Erin, then 1/2 the time we’d give a penny to Alice, 1/6 of the time we’d give it to Carol, and 1/3 of the time we’d start over. Alice would get 75% of the estate.