I'm trying to calculate and plot dice roll probability but am having some issues.
I want to find out the probability of rolling 4 or more 5s or 6s on n 6 sided dice. This is because in this game, should a player roll 4 or more 5s or 6s, they can role a second set of dice for a special effect.
I have seen that the formula from Bernoulli's trials:
$$P(r)=nCr*p^r*q^{n−r}$$
Can be used for the probability of $r$ successes in $n$ trials but it doesn't work for $r$ or greater success in $n$ trials.
When I used Bernoulli's trials I found that at $12$ dice or more trials the probability of rolling $4$ 5s or 6s started to go down which seemed strange to me. As such I assumed this was the wrong way to calculate
I have also tried to use a cumulative binomial function in Excel which may have given me the answer once I used 1-binim.dist() giving a 99% chance of 4 or greater 5s or 6s at 31 dice rolls. Again this seems high to me and I am not confident in the maths behind the excel formula.
Could someone please show me the maths for a cumulative binomial distribution or guide me in the right direction as to what it is I should actually be trying to calculate here.
Would this just be
$$P^k*(1-p)^n-k$$
And I just have a straight line graph of probability?