Games of Chance
There are many unpredictable, i.e., can't be justified external events put into unforeseen solutions. There exist some games that include an element of chance 1.
Example: Backgammon is typical game that combine luck and skill. Dice are rolled at the beginning of a Player's turn to determine the legal moves.
A game tree in backgammon must include chance nodes in addition to MAX and MIN nodes. These chance nodes are shown as circles.
The branches leading from each chance node denotes the possible dice rolls, and each is labelled with the roll and chance that it will occur.
The six doubles (1-1 through b-b) have a 1/36 chance of coming up, the other is district rolls a 1/15 chance each.
Only expected value can be calculated which leads to generalize the minimax value for deterministic games to an expectiminimax value for games with chance nodes.
Chance nodes are evaluated by taking weighted average values resulting from all possible dice rolls i.e.,
Successor function for a chance node a simply augments the state of n with each possible dice roll to produce each successor S.
P(S) - Probability that dice roll occurs.
