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.

Lets consider a sample size of 100000

The Above Historgrams confirms that B and C are proper Probability Distributions.

Find the probability that

(a) B + C < 1/2.

## [1] "The probability of B+C less than 1/2 is 0.1245"

(b) BC < 1/2.

## [1] "The probability of B*C less than 1/2 is 0.847"

(c) |B − C| < 1/2.

## [1] "The probability of |B-C| be less than 1/2 is 0.75102"