Probability theory and statistics

[Pouyan]

Homework Statement

The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function
f_X = (125-x)/450 , 95≤x≤125

How big is the probability of eight different runners, whose times are independent after 100 minutes:

a) Everyone has scored?
b) nobody has scored?

Homework Equations

I do know the distribution function is :

F_X(t) = P(X≤t) = ∫ (125-x)/450 * dx for x between 95 and t and we have
(250t - t² -14725)/900

The Attempt at a Solution

The right solution is :
[/B]
n=8 and from X₁ to X_n, we can describe independent stochastic variables, the time when the last one came into target:
Z=max(X₁...X_n)
And the time the first hit the finish line:
Y= min(X₁...X_n)
Further:
F_z(t) = P(Z<t)= P(max(X₁,...X_n)<t)= F_X1(t)...F_Xn(t)

in the same way:
F_Y(t)=1-(1-F_x1(t))...(1-F_Xn(t))
All in goal after 100 minutes: Z≤100
P(Z<100) = [F_X(100)]⁸ = (11/36) ⁸

None in goal after 100 minutes: Y> 100
P(Y>100) = 1-FY(100)= (1-F_x(100))⁸ = (25/36)⁸

How do I think :
I interpreted "The time everyone has scored after 100 minutes" as P (X> 100)
And tried with P(100<X<125) = 1-Fx(100) = 0.69
Then 0.69⁸ = 0.051
I know it's wrong, but this sentence "After 100 minutes ..." made me dizzy!

Why do we have to use max and min in this case?!

 

[mfb]

The two approaches look identical to me and the answers agree apart from rounding errors.