$X_1, \dots, X_7$ and $Y_1, \dots, Y_8$ are $\text{iid}$ with $N(\mu, \sigma)$ distribution.
I want to calculate a probability: $P(X_1<Y_{(8)})$, where $Y_{(8)}$ is $\max(Y_1, \dots, Y_8)$. I don' t know if I should search for a distribution of $X_1-Y_{(8)}$ and if so, I do not know how.
I will be grateful for any help in this task.
I'll do this by finding the probability of the complementary event $\big[ X_1\ge Y_{(8)} \big].$ Then you can subtract that from $1.$
First the hard way: $$ \require{cancel} \xcancel{ \begin{align} & \text{The answer is the same regardless of the values} \\ & \text{of $\mu$ and $\sigma^2,$ so I will do this using $\mu=0,$ $\sigma^2=1.$} \\[8pt] & \Pr(X_1 \ge Y_{(8)}) \\[8pt] = {} & \operatorname E( \Pr(X_1\ge Y_{(8)} \mid X_1)) \\[8pt] = {} & \operatorname E(\Pr(\text{all of } Y_1,\ldots,Y_8 \text{ are} \le X_1\mid X_1)) \\[8pt] = {} & \operatorname E((\Phi(X_1))^8) \\[8pt] = {} & \int_{-\infty}^{+\infty} (\Phi(x))^8 \varphi(x)\,du \\[8pt] = {} & \int_0^1 u^8\, du = \frac 1 9. \\ \phantom{+} \end{align} } $$ All correct, but here's the easy way: $$ \Pr\left( \begin{array}{l} \text{among nine iid random} \\ \text{variables } X_1,Y_1\ldots,Y_8, \\ \text{the largest is } X_1 \end{array} \right) = \frac 1 9 $$ by symmetry.