
BGonline.org Forums
The Backgammon Solution
Posted By: Casper Van der Tak
Date: Monday, 17 October 2016, at 7:03 a.m.
In Response To: The Backgammon Solution (Rick Janowski)
Maybe the following example can be helpful.
If you assume that the lower amount is drawn from a uniform, continuous distribution between 0 and 10,000, than the higher amount is drawn from a uniform, continuous distribution between 0 and 20,000. Both probability density functions will be horizontal lines at 1/10000 respectively 1/20000 (so that the integral sums to 1). If your envelop contains 11000, you know it is the higher amount (not very interesting). However, if you draw 2000, that is twice as likely the lower amount, so your incentive to and likely gain from the switch is even increased.
The amount in the envelop gives information about the relative probabilities of it being the lower or higher amount. The problem is that in order to use that information, you need to have some advance idea (prior) about the probability distribution over the different amounts, which you can then update on the basis of the actual observation. Here there are many possible priors, and you don't know which one is the right one.
My earlier 20 million example was a bit tongue in cheek, but also makes sense  unless people are really, really rich, they are much more likely to put a total of 30 million rather than 60 million in envelopes.
Look up Bayes for more information.

BGonline.org Forums is maintained by Stick with WebBBS 5.12.