I came up with the following while pondering the various probability puzzles of recent weeks, and I found it clarified some of my confusion
about the issues, so I thought I’d post it here to see if anyone else liked it:
Consider an experiment in which we toss a coin, to chose whether a person is placed into a one room hotel or duplicated and placed
into a two room hotel. For each resulting instance of the person, we repeat the procedure. And so forth, repeatedly. The graph of
this would be a tree in which the persons were edges and the hotels nodes. Each layer of the tree (each generation) would have equal
numbers of 1-nodes and 2-nodes (on average, when numerous). So each layer would have 1.5 times as many outgoing edges as incoming,
with 2⁄3 of the outgoing being from 2-nodes. If we pick a path away from the root, representing the person’s future, in each layer
we are going to have an even chance of arriving at a 1- or 2- node, so our future will contain equal numbers of 1- and 2- hotels. If we
pick a path towards the root, representing the person’s past, in each layer we have a 2⁄3 chance of arriving at a 2-node,
meaning that our past contained twice as many 2-hotels as 1-hotels.
I came up with the following while pondering the various probability puzzles of recent weeks, and I found it clarified some of my confusion about the issues, so I thought I’d post it here to see if anyone else liked it:
Consider an experiment in which we toss a coin, to chose whether a person is placed into a one room hotel or duplicated and placed into a two room hotel. For each resulting instance of the person, we repeat the procedure. And so forth, repeatedly. The graph of this would be a tree in which the persons were edges and the hotels nodes. Each layer of the tree (each generation) would have equal numbers of 1-nodes and 2-nodes (on average, when numerous). So each layer would have 1.5 times as many outgoing edges as incoming, with 2⁄3 of the outgoing being from 2-nodes. If we pick a path away from the root, representing the person’s future, in each layer we are going to have an even chance of arriving at a 1- or 2- node, so our future will contain equal numbers of 1- and 2- hotels. If we pick a path towards the root, representing the person’s past, in each layer we have a 2⁄3 chance of arriving at a 2-node, meaning that our past contained twice as many 2-hotels as 1-hotels.