## Questions on binomial distribution probability

Hello all

I've been working on this problem for a little while now but I don't
know if I am doing it right.

I am tying to use MS Excel to calculated the possibility of rolling
sets of dices for a game call Warhammer.

Basically the problem is this. I need to roll x amount of dices with
a fix possibility, then discard the fail result and re-roll the
successful roll with a different possibility, repeat and get n or
more dice remaining.

For example. I will roll 10 dices and keep all the dice that turned
up 4,5 or 6. Then roll the dices that I kept and keep all the dice
that turns up 5,6. Then roll the dices I kept a third time and keep
the once that turned up 6. What is the probability that 2 or more
dice remain at the end?

The reason I am doing this in excel is so that I can change the trial
size, probability and number of success. So I could be rolling 15 or
30 dice in the beginning, or I am keeping 3,4,5 or 6 instead of 5 or
6, or I want 4 dice remaining in the end, etc. Each stage is different.

So far I've been taking the cumulative binomial distribution
probability of the first dice roll with trial size 10, then multiply
it with the cumulative binomial distribution probability of the second
with trial size 10 and third dice roll with trial size 10. Then I realized,
I started out with 10 dices but each time I roll I may lose dice. I don't
know how to figure out the trail size for the second and third roll and If I
don't know the trail size I have no way of calculating the probability.

My questions are

1) Am I right in thinking that I can find the answer by calculating
the three rolls and multiply them together?
2) Am I using the right formula? (cumulative binomial distribution)?
3) If I am using the right formula, how can I found out the trial
size for the second and third roll?
4) Is there a better formula for calculating this?

Thanks all