In the "Not-always-lying politician" problem, what's the probability that a particular sentence is true?

Unless this politician is a terrific planner, I think it's safe to assume that he doesn't look at all 144 combinations and choose a random one.  Instead, a better way of choosing is for each statement, choosing true or false, unless the last one was false, in which case the statement must be true. 

My take is the equivalent of calculating according to each student's choice in "class pass" as opposed to calculating according to the total number of combinations.

So, the chances are 50-50 that the first statement is true.  The second statement is 75% likely to be true, because half of the time it must be true.  Likewise, each statement's probability of being true is 1-x/2, where x is the probability of the last statement.

So for all ten statements:

1  .5
2  .75
3  .625
4  .6875
5  .65625
6  .671875
7  .6640625
8  .66796875
9  .666015625
10 .6669921875

The infinitieth statement would have a probability of 2/3.  I'm making the assumption here that the politician forgot his 0th statement and his conscience isn't bothering him about it.

The average probability of those ten statements is .65556640625.

Although I found each statement's probability, I'm not sure what the function is for the nth statement.

Posted by Tristan on 2004-07-02 15:37:22