Three tanks A B and C are engaged in a battle Tank A when it

Three tanks A, B, and C are engaged in a battle. Tank A, when it fire it hits its target probability hits its target with hit probabilit and hit probability Initially (in the first period), B and C fire A and A fires at B. Once one is hit, the remaining tanks a at each other. The battle ends when there is one or left tank transition matrix for this game is (the states are the subsets of 22.

Solution

E(ABC)= a E(AC)= b E(BC)= c

From state AC how the game can be finished

either A dies or C dies or both die=5/12 + 1/12 + 1/12= 7/12

from state BC how the game finishes

either B dies or C dies or both die= 5/18 + 2/18 + 1/18= 8/18

from state ABC how game finishes

from state AC with prob=7/12 hence 5/18 * 7/12

from state BC with prob =8/18 hence 4/18 * 8/18

from C with prob =4/18

hence game finishes with prob

p=5/18 * p + 5/18 * 7/12 + 4/18 * 8/18 + 4/18

hence expectation= 1/p =1.495

b)

tank B winning

p=5/18 * p + 5/18 * 0 + 4/18 * 4/18

13/18 p=16/18*18

p=16/(13*18)=8/117

tank A winning

p=5/18 p +5/18 * 5/12

13/18 p = 5/18 * 5/12

p=5/13 * 5/12=25/156

tank C winning

p=5/18 * p + 5/18 * 1/12 +4/18 * 2/18 + 4/18

13/18 p =175/ 18*36

p=175/(13*36)=175/468