Math Help - Probability in an RPG

Hello Mathhelpforum!
Okay here's my problem.
In a game I am given the choice of two arrows:
1. Normal Arrows - 3gp (cost) - 1 damage (for each successful hit)
2. Iron Arrows - 5gp (cost) - 2 damage (for each successful hit)
As a character, I have a certain chance of hitting with every arrow I fire (which is the same for either arrow), and this chance increases as my character develops. What I would like to know is; What chance do I have to have of making a successful hit for it to be worth me purchasing the 'Iron Arrows' as appose to the ‘Normal Arrows’ (because whenever you miss you waste 2gp more with the ‘Iron Arrows’).
Thanks a lot =)

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Ops, I posted this in the wrong thread. Sorry, this should be in 'Las Vegas Math'

Tough Problem. It seems simple enough, but you have a currency problem. Unfortunately, the repair of the currency problem is up to you. What is a damage point worth to you in gp? It must be greater than 2, I think.

Originally Posted by Obsidantion

Hello Mathhelpforum!
Okay here's my problem.
In a game I am given the choice of two arrows:
1. Normal Arrows - 3gp (cost) - 1 damage (for each successful hit)
2. Iron Arrows - 5gp (cost) - 2 damage (for each successful hit)
As a character, I have a certain chance of hitting with every arrow I fire (which is the same for either arrow), and this chance increases as my character develops. What I would like to know is; What chance do I have to have of making a successful hit for it to be worth me purchasing the 'Iron Arrows' as appose to the ‘Normal Arrows’ (because whenever you miss you waste 2gp more with the ‘Iron Arrows’).
Thanks a lot =)

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Ops, I posted this in the wrong thread. Sorry, this should be in 'Las Vegas Math'

In order to make a decision you need to know the value of the damage
you do, that is you need to know the utility (expressed in gp's is easiest)
of a point of damage. In a game system without easy resurection the
expected value of each point of damage done can be enourmous to the
charater (and possibly negligable to the player).

However you don't give enough information for us to solve the decision
problem.

RonL

Thanks for the thoughts guys but I think I know the answer (I did lay it out pretty vaguely):
The 'Iron Arrows' will always be worth buying so long as my successful hit rate is above 0 because they have a larger Damage to Cost ratio.

Originally Posted by Obsidantion

Thanks for the thoughts guys but I think I know the answer (I did lay it out pretty vaguely):
The 'Iron Arrows' will always be worth buying so long as my successful hit rate is above 0 because they have a larger Damage to Cost ratio.

Why is damage to cost ratio important?

What opportunities have you forgone by spending money on iron arrows?

RonL

If normal arrows were 1 gp each the damage to cost ration would be better than iron arrows, but would you actually use them in combat where you could loose loose all your armour etc to a competitor with iron arrows? That would be a false economy.

However if you were only training your character against safe targets in order to level up your character then normal arrows might be OK. However, the leveling up process would take much longer and that time might be better spent making goods etc to earn more money and then using more expensive arrows to power level your combat ability.

I have a new opinion. Let's examine a maximum of 2 damage.

p = Pr(hit)
q = 1-p = Pr(miss)

Damage Expectation

Iron: Cost is 5

Normal Cost is 6

The choice, then, is do you want to pay 5 or 6 to have an expectation of 2p? Buy the iron arrows, unless there is some advantage to doing some damage rather then no damage.

Originally Posted by CaptainBlack

Why is damage to cost ratio important?

What opportunities have you forgone by spending money on iron arrows?

RonL

I may have given the wrong impression here. If I were you I would buy
iron arrows. But I would not seek a pseudo-scientific justification for it.

The point I am trying to make is that things like damage to cost ratios
are not the decision variable relevant to the problem. The real decision
variable is the probability of surviving an adventure (or something similar
your objective is not the same as that of your character). In practice this
is too complex to model adequately so we resort to other methods.

Now human beings come equipped by evolution with mechanisms for making
complex decisions under uncertainty and with incomplete information, so
unless this is a homework problem requiring you to use some false objective
method, let your gut decide. After all you let it decide much more serious
matters every day.

RonL

Some excellent points put out in this thread, especially that of how the use of the arrows in a game is much more dynamic then to just consider cost and damage (‘time taken to get exp if that is shortened with a more powerful arrow’, ‘damage wasted if a target has only 1 hp left with 2 hp damage’ etc also matter).
However, this question is no longer practical as I have discovered that the iron arrow does not take double the damage of the normal arrow and I have been advised by an experienced player to use the normal.
Although, I would like to continue contemplating this question under the restrictions that the only things that ‘matter’ are the damage that you do (taking into account the chance of you missing) and the cost of the arrows you use to make that damage.
I'll rephrase the question to clarify: 'What is the lowest chance you must have of making a successful hit for it to be (probably) worth you purchasing the Iron Arrows instead of the Normal Arrows?'
My answer would be that the Iron Arrows are always more beneficial provided your chance of a successful hit were above 0% because the damage made costs less respectively, but I'm not sure so this is why I have left the question open for discussion (I hope this makes sense).

Thank you for your input CaptainBlack, TKHunny and kev p

Originally Posted by Obsidantion

Some excellent points put out in this thread, especially that of how the use of the arrows in a game is much more dynamic then to just consider cost and damage (‘time taken to get exp if that is shortened with a more powerful arrow’, ‘damage wasted if a target has only 1 hp left with 2 hp damage’ etc also matter).
However, this question is no longer practical as I have discovered that the iron arrow does not take double the damage of the normal arrow and I have been advised by an experienced player to use the normal.
Although, I would like to continue contemplating this question under the restrictions that the only things that ‘matter’ are the damage that you do (taking into account the chance of you missing) and the cost of the arrows you use to make that damage.
I'll rephrase the question to clarify: 'What is the lowest chance you must have of making a successful hit for it to be (probably) worth you purchasing the Iron Arrows instead of the Normal Arrows?'
My answer would be that the Iron Arrows are always more beneficial provided your chance of a successful hit were above 0% because the damage made costs less respectively, but I'm not sure so this is why I have left the question open for discussion (I hope this makes sense).

Thank you for your input CaptainBlack, TKHunny and kev p

My only thought is that there may be something out there that requires the iron arrow to punch through (scales, armor, etc.) or that might take extra damage from iron weapons (such as faerie creatures or certain kinds of elementals, depending on what kind of RPG you are using.) Certainly when I was "DMing" I took such factors into account. (But then a computer game might not. Of course, I let my players' iron rust if they stayed wet for too long, too. )

-Dan