1. ## Re: Game probability problem

OK thanks. And is it otherwise correct?

2. ## Re: Game probability problem

Originally Posted by BGM
Let be the number of attackers survived just before the -th battle (if exist), so it is including those singleton attacker stayed in the territory. And is the initial number of the attackers. Note that the actual number of attackers entering the the -th battle is .

Let be the number of defenders entering the -th battle.

Let be the probability mass function of the number of attackers left after a battle given that the number of attackers and defenders entering the battle are

Therefore the probability of winning is

Sorry if I count it wrong
Please help, I need to know if my 3-battle solution is correct so I can generalize. (See post #29)

Furthermore, could you show how to find the probability for the 2-battle scenario of there being attackers surviving and placed on the last conquered territory at the end (in terms of the pm functions we already know how to calculate), for where is the number of attackers that originally started (maximum is because the first battle involves attackers, the second involves maximum attackers and if these attackers all win there will be surviving attackers on the last conquered territory).

3. ## Re: Game probability problem

I think #29 is alright (at least the same as my idea but maybe I am wrong). You may try to verify it by simulation.

For the latter problem you want to ask if there is exactly attackers survive and placed on the last territory.

Note that in the formula means we consider all the scenario with the attackers from to so that you are winning the last battle. (Actually is the you considered here).

Therefore you just need to modify it as

Note that you require survived after the second last battle.

4. ## Re: Game probability problem

Thanks. So what are the two summation bounds for the 3-battle case?

But what are the summation bounds ...

5. ## Re: Game probability problem

Actually there is nothing special/mysterious. Just need to include all the cases, and ensure that it is possible to led to survivors at last.

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