Need help with a game design.

[Geikamir]

Hello all.

I am working on a game idea that I have but I am stuck on a part where I need to know the probability of an outcome so that I can properly balance the game before actual playtesting. I have tried to do some learning of probability formulas from Google and YouTube, but I'm having trouble making it applicable to this particular problem. I was hoping someone here might be able to help me understand what formula to use so that I can continue designing and testing this idea.

Ok, so on to the scenario:

There is a deck of cards with 80 total cards. 26 of those cards are 'Stars' (more in those below**). The remaining 54 cards 2-10 in six different suits.


During each 'set up' players keep drawing from the deck until the total value of 4 or more is reached (possibly 5, depending on what the averages wind up being) and then they stop. This is called the 'break point'. I'm wanting to know what the average 'set up' value is.




**The Stars are what throw a bit of a kink in the math. If only a single Star is a part of the 'set up' its value is just 1. However, as more are drawn before hitting the 'break point' (currently 4, as mentioned above) their total value increases in triangular fashion. 1/3/6.


So, as an example: If two 3's are drawn the total 'set up' value is 6 because the first drawn 3 didn't meet the break point threshold of 4+ and then a second card was drawn, which happened to be a 3.

Another example is: If two Stars are drawn first and then a 10 is drawn, the total encounter value is 13. That's because the first Star is value 1, the second Star turns their total value into 3, and then a third card is drawn (because they are under the break point) which happens to be 10 in this scenario.

With that information, what is the average value of a 'set up' drawn from the deck with the break point being 4? What about if it was 5? What is a good formula to use to figure this type of problem out?


Thanks a lot for your time! All help is greatly appreciated!

 

[ImperfectSphere]

I have tried to do some learning of probability formulas from Google and YouTube, but I'm having trouble making it applicable to this particular problem.

Could you tell us how well versed you are with this topic? Because the videos on Youtube are fairly descriptive

 

[Geikamir]

Could you tell us how well versed you are with this topic? Because the videos on Youtube are fairly descriptive

Getting the odds of drawing any singular particular card I've learned. That part seems easy if I'm doing it correctly. Divide the amount of that particular card in the deck (for example: there are six 2's. So 6/80= 7.5% chance to draw a 2).

I think the way to figure out the odds to draw anything 5+ is to add up all of those cards and divide by the deck size.

So,I think the chance to only draw a single card is about 45% and I think that average card value with a single card is 7.5.

I start getting really bogged down on figuring out the value if the 2 or 3 cards are needed. The general idea seems to take the number of cards left that can cause exactly 2 cards and no more then divide by the remaining deck size (79 now at card 2).

However: The Stars have a changing value and that is totalling messing me up. HowntonI account for that?

Also, once I have a % chance for each number of cards to come out (1, 2, or 3) how do I find the expected value of each. And then how to I find the total average value?

I've tried to just logic it out by using a bunch of janky steps to get to these answers, but I've thrown quite a lot of rounding and guessing in there. My current figure is that the average value of an average 'set up' is ~8. I think the odds of drawing exactly 2 cards is ~25% and the odds of drawing 3 cards is ~30% (4 cards is impossible with the current 4 or below threshold).

It took longer than I want to admit piecing together those numbers and I'm very confident that they aren't exactly accurate. Might not even be close.