Wikus van de Merwe in his alien battle robosuit and Colonel

Wikus van de Merwe, in his alien battle robo-suit, and Colonel Koobus are taking turns shooting at each other, until the first time somebody is hit. (the series may look like \"....Wikus misses, Koobus misses, Wikus misses, Koobus hits\": at this moment the game stops with Koobus as a winner). At each shot Wikus can hit his opponent with the probability of 1/3, Koobus - with the probability of 1/2. All shots are independent. Show that that game can not last forever. Show that, if Wikus shoots first (that is, Wikus starts the game), then he has the probability 1/2 to be the winner in the end; Show that, if Wikus shoots second (i.e., Koobus starts the game), then the probability of Wikus to be the final winner drops to 1/4.

Solution

The game will not last forever as there is a certain result that ineveitable.

If Wikus shoots first then he has two chances of hitting hence the probability is 50% or 1/2

If Wikus shoots second then he has only one chance of hitting hence the probability is half of the probability at two chances that is 25% or 1/4