San josé State University:
www.applet-magic.com/samplemin.htm
it can be read:
******** If p(x) is the probability density function for a random variable x, let P(x) be the cumulative probability function; i.e.,
P(x) = Intg(-infinity, x).p(z)dz
The probability that the minimum of a sample of size n is x is given by
Prob.(min size n sample) = n[1- P(x)]^(n-1) * p(x)
*********************************************
My comment
Suppose, for example,
Uniform [0, 1] Distribution, P(x) = x, p(x) =1
Prob.(min size n sample) = 21[1- x]^20 * 1
If x=0.05 then results 7.528,
This is, no doubt, a remarkable thing (!!!). The Applet is so magic that it not available yet. Please, remove it!
The writer did intend to say:
____Prob (min <=x) = 1- (1-F(x))^n
____1- (1-0.05)^21=0.659438______
Simulation (data 8E6 items) gave 0.660, 0.660.
Prob (absolute deviation > 0.0005 | N=8E6) <=
2*EXP(-2*8E6*0.25*1E-6) = 2*EXP(-4) = 0.037= 3.7%
(According to Dvorestky - Kiefer - Wolfowitz inequality, states).
REM "ckontas"
        CLS
        DEFDBL A-Z
        DIM x(22)
        INPUT " x=  "; x0
        p = 1 - (1 - x0) ^ 21
        PRINT USING "#.### "; p
        all = 8000000
        RANDOMIZE TIMER
        FOR j = 1 TO all
        LOCATE 4, 4: PRINT USING "########"; all - j
        minor = 8
        FOR i = 1 TO 21
        x = RND
        IF x < minor THEN minor = x
        NEXT i
        IF minor > x0 THEN GOTO 10
        okk = okk + 1
10      NEXT j: LOCATE 10, 10
        PRINT USING "#.### "; okk / all
        END