This problem reminds me the
German Tank Problem, in which allies had to determine the number of tanks available to germans by observing a few german tanks which had a progressive number on it.
Let me restate the rules before going on.
- You can bet a fixed X amount;
- you have to decide beforehand the number of years Y for your investment;
- if the animal is still alive after Y years, you get X*2^Y back, for a net winning of X*(2^Y-1) (is that correct?);
- if the animal dies before Y years, you lose X;
- you know that the animal is 20 years old;
- you have no knowledge about life expectancies of animals in the world.
Regarding the last point above, I interpret as that you don't have access on any data and so you don't know the "correct" distribution. However, you are allowed to use some personal prior (otherwise the problem can't be addressed at all in terms of probabilities), since you know that a random animal is much more likely to live for some decades rather than many centuries.
Say L is the age at which the animal will die. The probability of you meeting it while it being 20 year old is 1/L (you might have met it at any age in its life) if L>=20 and 0 otherwise of course. So, using Bayes' rule:
P(L | we met it at 20) ~ P(we met it at 20 | L) * P_0(L) = 1/L * P_0(L) for L>=20
where with ~ we intend "proportional to" (we have to normalize the result afterward) and P_0(L) is our prior probability of a random animal dying at L years.
As we can see, we need to build a prior. Say that we have a uniform prior over some [0,M] range. We can see that our posterior over L is 0 if L<20 and then decreases like 1/L to L=M, after which is 0 again. If you bet on Y+20, you win with probability P(L>=Y+20). This probability decreases about logarithmically with Y (being the complement of a sum of 1/L terms), while the payout increases exponentially with Y. So, with a uniform prior, you should go all-in and bet the maximum age you think the animal can live.
Of course the result is very dependent on the prior you use. If you use some exponentially decaying prior (or if you add some utility function, since not every dollar is worth the same) you might arrive to very different conclusions. However, I guess this is how the problem should be approached.
I don't think the link David provided is any relevant. In the sunrise problem you don't know that there will be a time in which the sun won't rise anymore. Here we know that the animal will die. In the sunrise problem is even conceivable that the sun might not rise some days, but rising back afterwards (that's the reason why they treat the problem as binomial, with a fixed p probability of happening regardless what happened before).
