a little war game
One day, 2 armies meet each other and begin fighting. The soldiers in the 2 armies are so equally strong, that if one soldier fought another soldier, their power would cancel out and they would both die (yeah, I know it's pretty dark). How the battle goes, however, is quite interesting. Each army sends one soldier to battle one another and waits until this 1 on 1 skirmish finishes to repeat this process again. This goes on until one army runs out of soldiers to send, thus ending the battle and making the army with soldiers left, victorious.
Now let's say the sizes of the 2 armies are 5 soldiers and 8 soldiers and that you were a random soldier in one of the 2 armies. The chances of you surviving the battle are . What is ?
The answer is 16.
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1 solution
First, we should answer the question "how many 1 on 1 skirmishes will happen in this battle?" The answer to this is 5 since army 1 with 5 troops has the smallest amount of soldiers compared to 8 with army 2. So, army 1 will send out their 5 soldiers but they will all die since it is not enough to eliminate army 2 completely. But since army 2 also has to send a soldier to battle with the enemy and die, army 2 will also send and lose 5 men.
The number of soldiers remaining and surviving in army 2 will be 8 − 5 = 3 , 3 soldiers. The number of soldiers in both armies are 5 + 8 = 1 3 , 13, and out of these 13, only 3 survived. Therefore, the chance of any soldier (including you) is 3 / 1 3 . And to get our answer of 16 we add. 3 + 1 3 = 1 6