lpm wrote: ↑Fri Sep 01, 2023 1:15 pm

If you look again at your own wording:

60% chance Nothing much happens in next year - Biden wins

20% chance Trump has health crisis - Biden wins

20% chance Biden has health crisis - Trump has a chance

line one is a certain outcome

line two is a certain outcome

line three is where we have placed the missing element, stating in black and white only that "Trump has a chance" rather than "Trump wins".

Yes, and that I've changed to Trump wins for simplicity (because Trump having a better chance includes outcomes where Biden wins, and so there's overlap between the scenarios). Change it back to Trump has a chance throughout, and it's technically more complicated.

The strawman scenarios as originally listed are:

1. 60% chance: Nothing much happens resulting in a Biden win.

2. 20% chance: Trump has a health crisis resulting in a Biden win.

This includes scenarios where both have health crises, but Biden stays in the race, and wins.

3. 20% chance: Biden has a health crisis resulting in Trump having a better chance

This includes scenarios where:

Both have health crises, with Biden withdrawing from the race, and Trump pushes through it to have a better chance against the backup Democrat. Both have health crises, with both staying in, but Trump's odds improve over Biden's.

Only Biden has a health crisis, resulting in improved odds for Trump against the either Biden or the backup Democrat.

Missing scenarios, excluded from your enumeration of outcomes, and so not part of the normalization:

Biden has a health crisis, Trump doesn't, and the outpouring of sympathy and Trump's ugly reaction to it improves Biden's chances.

Biden has a health crisis, withdraws, and the replacement Democrat has an better chance of beating Trump than Biden (whether due to the replacement's greater electability, sympathy or Trump's ugly reaction to Biden's health crisis)

Both have a health crisis, and withdraw.

So under this detailed enumeration of scenarios, comparing the strawman odds including 1,2 and 3, with the odds including 1 and 3 only, the odds of Trump having a chance are 1 to 4 (20% to 80%) and 1 to 3 (20% to 60%). But the odds of a Biden win are not 3 to 4 and 2 to 3.

Note that probabilities are not assigned to the chances of "Trump having a health crisis" and "Biden having a health crisis". You can calculate them under various assumptions about the missing scenarios if you like.