I play a tabletop game and we roll dice for game outcomes. The dice have 8 sides and a certain result (lets call it a success) appears on 2 sides, so 25% of success.
If I roll 4 dice, the chance of having at least one success is about 70% (binomial distribution for 4 dice).
Say I'm rolling 4 dice, and then I'm rolling another 4 dice. Statistically, this is the same as rolling 8 dice, right? Therefore, I can apply the binomial distribution to 8 dice. The chance of 5 or more successes in 8 dice is about 3%.
Say I roll 4 dice and get 4 successes. I can conclude that the chance of having at least one other success in the next 4 dice is about 3%.
However, a lot of other players disagree and think that it is not 3% but 70%, because I'm rolling 4 dice again, and it doesn't matter what I rolled before.
I would appreciate if you could either confirm I am right, or explain to me why I'm wrong. thanks!