Question

8.7

Naval Battle Analysis

In the game Axis & Allies, the outcome of a two-sided naval battle is decided by repeated rolling of dice. Until all ships on at least one side are destroyed, each side rolls one (six-sided) die for each of its existing ships. The die rolls determine casualties inflicted on the opponent; these casualties are removed from play and cannot fire (roll) in subsequent rounds. There are two types of ships: battleships and destroyers. For a battleship, a die roll of four or lower is scored as a “hit.” For a destroyer, a die roll of three or lower is scored as a “hit.”
It takes two hits (not necessarily in the same round) to destroy a battleship and only one hit to destroy a destroyer (side note: battleships are twice as expensive as destroyers). The defender gets to decide to which ship to allocate the hit; we assumethedefenderchoosesintelligently.Iftwodestroyersengageabattleship in a naval battle, what is the probability that the destroyers win? How about the battleship? [Hint: You will need to raise a matrix to a large power.]

Answer 1

Probablity of a destroyer taking a hit = 1/3

Probablity of a destroyer not taking a hit = 1-1/3 = 2/3

Probability of a battleship taking a hit = 1/4

Probability of a battleship not taking a hit = 1-1/4 = 3/4

Probability of the destroyer winning = (2/3)*(1/4)*(2/3)*(1/4) = 0.02777777777 (Answer)

Probability of the battleship winning = (1/3)*(3/4) + (1/3)*(1/4) = 0.33333333333 (Answer)