# 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

n <- 100000
B <- runif(n)
C <- runif(n)
b <- as.data.frame(runif(n, min=0, max=1))
c <- as.data.frame(runif(n, min=0, max=1))

hist(b[,1], prob=TRUE, ylim=c(0,2), main = 'Histogram of B',col = "orange")

hist(c[,1], prob=TRUE, ylim=c(0,2), main = 'Histogram of C',col = "light green")

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

# Find the probability that

### (a) B + C < 1/2.

print(paste("The probability of B+C less than 1/2 is",(sum((B+C) < .5)/n)))
## [1] "The probability of B+C less than 1/2 is 0.1245"

### (b) BC < 1/2.

print(paste("The probability of B*C less than 1/2 is",(sum((B*C) < .5)/n)))
## [1] "The probability of B*C less than 1/2 is 0.847"

### (c) |B − C| < 1/2.

print(paste("The probability of |B-C| be less than 1/2 is",(sum(abs((B-C)) < .5)/n)))
## [1] "The probability of |B-C| be less than 1/2 is 0.75102"

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

max <- 0

for(i in 1:n){
if(max(B[i], C[i]) < 0.5){
max = max + 1
max
}
}
max <- max/n
print(paste("The probability of max{B,C} less than 1/2 is",max))
## [1] "The probability of max{B,C} less than 1/2 is 0.2488"

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

min <- 0

for(i in 1:n){
if(min(B[i], C[i]) < 0.5){
min = min + 1
min
}
}
min <- min/n
print(paste("The probability of min{B,C} less than 1/2 is",min))
## [1] "The probability of min{B,C} less than 1/2 is 0.74821"