How to translate $x$ when moving from uniform distribution to order statistics with beta distribution? Let $Y_{i}$ be the number of days until package $i\in[1,7]$ comes (independent). We know that $Y_{i}\sim Uni\left(0,10\right)$. Find the probability so it will take more than 8 days for the last package to come.
So I defined $Y_{(i)}$ be order statistics. So we get $Y_{(7)}\sim Beta\left(7,1\right)$. I would like to calculate $P\left(Y_{(7)}>8\right)$ but $x$ in $P(X>x)$ should be $x\in[0,1]$ by def of beta distribution. So how I translate $8$?
 A: In the following, I assume that $Y_i$ is a continuous random variable (in particular, the 'number of days' does not have to be an integer).

I assume the result you have used is that if $X_1,\dots,X_n$ is a random sample drawn from $U(0,1)$ then the $k$-th order statistic $X_{(k)}$ follows the distribution $\mathrm{Beta}(k,n+1-k)$ for all $k\in \{1,2,\dots,n\}$.
However, note that this result only holds for uniform distributions on the standard unit interval, whereas $Y_i\sim U(0,10)$. To solve this problem, define $Z_i:= Y_i/10$, then $Z_i\sim U(0,1)$. Clearly, the ordering of the random variables is preserved by division by $10$, so one also has that $Z_{(i)}=Y_{(i)}/10$. Then one can use the theorem you have used to conclude that $Z_{(7)}\sim \mathrm{Beta}(7,1)$. Therefore the result is
$$\mathbb{P}(Y_{(7)}>8)=\mathbb{P}(Z_{(7)}>0.8)=\int_{4/5}^1 \frac{\Gamma(7+1)}{\Gamma(7)\Gamma(1)}x^{7-1} (1-x)^{1-1}~dx=x^7\Big|_{4/5}^1=1-(4/5)^7.$$
A: Calling it the "number of days" seems to suggest it's an integer, i.e. you have a discrete uniform distribution. Moreover, the number of days is not with in the interval from $0$ to $1,$ so you don't have a beta distribution, although you could have a re-scaling of a beta distribution.
The probability that all packages arrive in no more than $8$ days is the $7\text{th}$ power of the probability that a particular package arrives in no more than $8$ days, so it is $(9/11)^7$ (since there are nine numbers in the set $\{0,1,2,3,4,5,6,7,8\}$ and $11$ in the set $\{0,1,2,3,4,5,6,7,8,9,10\}.$)
