binomial distribution and rolling dice 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!
 A: Unfortunately, you are wrong, and the other players are right. The chance of you getting at least 1 success out of 4 dice is still 70%, because previous events don't affect future events. 
Think about it this way - you have a coin that you know is fair (50/50). You flip it 1,000 times, and it lands on heads every time. What are the odds that it'll be a head the next time you flip it? It might seem like the coin has 'used' up all of its luck already in getting heads, but it's still 50/50 because previous outcomes don't affect future ones. Coins aren't affected by previous outcomes - otherwise, you'd need to get a brand new coin each time you wanted to flip a coin fairly. In the same manner, your 4 dice rolls aren't affected in any way by your previous rolls. 
In more mathematical terms (this isn't the proper notation but hopefully you get the gist)
$P(success = 1$ out of $4) = 0.70$
$P(success = 5$ out of $8) = 0.03$
However, you've already rolled 4 dice, so you need to account for that. The real probability you're looking at is:
$P(success = 5$ out of $8 | success = 4$ out of $4)$
which is equal to 0.7. 
(All of this assumes that your dice are actually fair - if they're rigged that's another story.)
