Let $n$ be the number of 6-sided dice thrown. Further, let $X =$ "Greatest thrown number".
What are the probabilities $P[X=k]$ for every $k=1,...,6$?
Obviously, $P[X=1] = \frac{1}{6^n}$.
I also know that $P[X=k] = P[X\leq k] - P[X\leq k-1]$. Therefore, $P[X=2] = P[X\leq 2] - P[X=1]$.
However, I don't know how to calculate $P[X\leq 2]$ for $n$ dice, although I'm pretty sure it involves the Binomial coefficient?
Thank you in advance!
Let n be the number of 6-sided dice thrown and $$k=1,...,6$$. X= "Greatest thrown number".
What are the probabilities $P[X=k]$ for every k=1,...,6?
let us consider the events:
$A_k=${getting a number less than k in the dice} and $X_i$ the result of $i_{th}$ dice
$$P[X=k]=P[X\leq k]-P[X \leq k-1]$$ $$P[X=k]=P[\bigcap (X_i \leq k)]-P[\bigcap (X_i \leq k-1)]$$ $$P[X=k]=\prod P[(X_i \leq k)]-\prod P[(X_i \leq k-1)]$$
$$P[X=k]=P[A_k]^n-P[A_{k-1}]^n$$
$$P[X=k]=(\frac{k}{6})^n-(\frac{k-1}{6})^n$$