I'm trying to work out if random variance in dice rolls is more likely to influence a given situation in a game rather than the overall expected values of those dice rolls being significant. The game is a common table-top miniature game, where one must roll certain dice in succession but only if you've previously scored a success.
To give frame to the question, let's assume the following:
- The initial number of dice to be rolled is 20
- If a roll is 'successful' then that die is used in the next round of rolling, and if it is 'unsuccessful' then it is removed
- There are 3 rounds of rolling
- Round 1 success is determined by a roll of 4, 5, or 6
- Round 2 success is determined by a roll of 3, 4, 5, or 6
- Round 3 success is determined by a roll of 5 or 6
Now the issue I have is that quite often people will look at the overall expected value of the dice rolling game and make assumptions based on this. In our scenario, with 20 dice and consecutive probabilities of success of 1/2, 2/3, and 1/3, the overall expected value of our final successful dice at the end of round 3 would be 2.22 (if not rounding to whole numbers).
However I'd imagine that the individual variance (if I'm using the correct word here) of any given set of dice rolls would play a far more important role than that final expected value given the relatively small sample space of only 20 initial dice.
So my question is two-fold:
- How many dice would you have to roll to be relatively certain of getting close to your expected value of final successes?
- Given the above example of rolling 20 dice, what certainty would you have in getting that expected value?
For argument's sake, let's say that I'd like to be 90% certain of getting within 1 either side of my expected value for question 1 - and feel free to use the above numbers to illustrate if it's easier.
Happy to answer any clarifications as required!
Thanks for your help!
EDIT FOR CLARIFICATION:
Let’s take my problem to an extreme. With the probabilities for success as above (50%, 66.6%, then 33.3%), if I were to roll 1 die my ‘expected’ outcome mathematically to get three successful rolls would be 0.11. However, by the end of the three rolls I will either have a value of 0 successes, or 1. Both are distant from my expected value.
Even if we increase the number of dice to 10, with a now increased expected value of final successes of 1.11, it’s plausible that I could end up with 6 successes overall, which again would be distant from my expected value.
However, if I increased the number of dice to 10,000,000, there’s a high probability that my final successes would be close to my expected value of 1,111,111 given the likelihood that many outlying rolls would not confer statistical significance.
So, at what ‘number of dice’ rolled does the probability of my final results falling within, say, one standard deviation on either side of my expected value, become 90%? How many dice must I roll before the chances of ‘randomness’ affecting my overall result is statistically reduced to less than 10% ie. I am 90% likely to achieve final successes within one standard deviation on either side of my expected value?
Hope that clarifies it somewhat!