There are $n$ symmetric dice. Calculate the probability that the maximal result is equal to $k$ for $k=1,2,3,4,5,6$.
First, the dice are symmetric so the chance to get any number is equal, so my $\Omega$ is symmetric. There are different dice, so I use ordered selections and two dice can show the same number so:
$$|\Omega| = 6^n$$
Now, I defined $A$ to be the max result equals to $K$, and I need to calculate $\frac {|A|}{6^n}$.
I don't know how to continue and would love for some help. Thank you!
Hint:
If $M_n$ denotes the maximum result by throwing $n$ fair dice then: $$P(M_n=k)=P(M_n\leq k)-P(M_n\leq k-1)$$ The terms on RHS are not difficult to find.
Observe that: $$\{M_n\leq k\}=\{D_1\leq k,\dots,D_n\leq k\}$$ where $D_i$ denotes the result of the $i$-th die.