Probability it takes more then 25 rolls of a six sided die to land on a 6 a total of seven times?

Question

Say you're rolling a die and you continue until you roll a $6$ a total of $7$ times, what is the probability that it takes more than $25$ rolls? The best answer I got was by using the cumulative binomial distribution, so $P(x>25) = 1-p(x<25) = .913$ but I'm not sure if that's right. I'm fairly certain that the expected number of rolls would be $42$, so my answer would make sense but I'm not sure if I went around it the right way.

Answer 1

It will take more than $25$ rolls to obtain $7$ sixes if and only if the number of sixes you obtain in $25$ throws is $6$ or fewer. The number $\ N\ $ of sixes you obtain in $25$ rolls follows a binomial distribution with parameters $25,\frac{1}{6}\ $. That is \begin{align} P(N\le 6)&=\sum_{i=0}^6 {25\choose i}\left(\frac{1}{6}\right)^i\left(\frac{5}{6}\right)^{n-i}\\ &\approx0.891 \end{align}