How to find $P_{\theta}\big[\text{max}(X_1, \cdots, X_{100}) > 30\big]$ where $X \sim \text{Normal}(\mu = \theta, var=20)$

Question

I have $X \sim \text{Normal}(\mu = \theta, var=20)$ I need to find $P_{\theta}\big[\operatorname{max}(X_1, \cdots, X_{100}) \big]> 30$

What I tried:

$P_\theta (X_1 > 100) = 1 - \Phi(\frac{100-\theta}{\sqrt{20}} ) $

but I can't proceed. Is there anyone to help me out?

Answer 1

Hint:

$$P_\theta \left[ \max(X_1, \ldots, X_{100})>30\right]=1-P_\theta \left[ \max(X_1, \ldots, X_{100})\le30\right]$$

Now think about if the maximum is less than or equal to something, what can we say about each $X_i$.

Remark about your attempt:

I don't think the event that $X_1 > 100$ is of interest.