You roll a dice up to three times. After each of the first two rolls you can choose to accept the number showing on the dice in dollars, or roll again. What is your expected winnings if you play optimally?

dicegame.hs

import Control.Probability

expected p = expectation (runProb p)

die = uniform [1..6] :: Bayes Double Integer

game n
  | n == 0    = die
  | otherwise = do
    x <- die
    if fromInteger x > expected next
      then return x
      else next
 where
  next = game (n-1)

{--------------------

>> printProb $ game 2
1    5.56%
2    5.56%
3    5.56%
4   16.67%
5   33.33%
6   33.33%

>> expected $ game 2
4.66666666666667

---------------------}