How to find probability after finding the CDF of a max?

Question

I have the pdf:

$$ f(x)=10x^9, 0 < x < 1$$

Assuming independence, let Y=max(X₁,...X₈).

Find the cdf for Y: I found this to be y⁸⁰

Now I have to find P(.9999 < Y < 1)

I thought this would be:
P(.9999 < Y < 1)=
P(.9999 < X₁ < 1)*...*P(.9999 < X₈ < 1)
=[P(.9999 < X1 < 1)]⁸
$$(\int_.9999^1 10x^9 dx)^8$$
but that turns out to be a very small number, so I think I did something wrong...

Answer 1

Hint:

• What is the probability $0 \le X_1 \le 0.9999$?
• What is the probability $0 \le X_i \le 0.9999$ for all $i$?
• What is the probability $0 \le Y \le 0.9999$?
• What is the probability $0.9999 \lt Y \lt 1$?