HW5

C <- runif(10000, min = 0, max = 1)
B <- runif(10000, min = 0, max = 1)
sum(B>0 && B<1 )

## [1] 1

sum(C>0 && C<1 )

## [1] 1

sum(B+C>0 && B+C<2 )

## [1] 1

so B and C are proper probablity distribuction

B+C <1/2

P = 1/21/21/2 = 0.125

BC<1/2

C <- runif(10000, min = 0, max = 1)
B <- runif(10000, min = 0, max = 1)
round((sum((B*C) < .5)/10000),2)

## [1] 0.85

|B-C|<1/2

C <- runif(10000, min = 0, max = 1)
B <- runif(10000, min = 0, max = 1)
round(sum(abs(B-C) < 0.5)/10000,2)

## [1] 0.75

max(B,C) <1/2

P(B)<1/2 and P(C) < 1/2 = 1/2*1/2 = 0.25

min(B,C) < 1/2

1- P(B)>1/2 and P(C)>1/2 =1-0.25 = 0.75