Choose independently two numbers B and C at random from the interval [0, 1] with uniform density. Prove that B and C are proper probability distributions. Note that the point (B,C) is then chosen at random in the unit square.

set.seed(123)
B <- runif(10000, 0, 1)
C <- runif(10000, 0, 1)

Find the probability that:

(a) \(B + C < 1/2\)

X <- B + C
sum(X < 1/2)/10000
## [1] 0.1217

(b) \(BC < 1/2\)

X <- B * C
sum(X < 1/2)/10000
## [1] 0.8562

(c) \(|B − C| < 1/2\)

X <- abs(B - C)
sum(X < 1/2)/10000
## [1] 0.747

(d) \(max\{B,C\} < 1/2\)

sum(pmax(B, C) < 1/2) / 10000
## [1] 0.2558

(e) \(min\{B,C\} < 1/2\)

sum(pmin(B, C) < 1/2) / 10000
## [1] 0.7597