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. Find the probability that

For uniform distribution http://www.r-tutor.com/elementary-statistics/probability-distributions/continuous-uniform-distribution

(a) B + C < 1/2.

# B + C < 1/2.

p = vector("numeric", 500)
for (i in 1:500) {
  B = runif(1, min=0, max=1) 
  
  C = runif(1, min=0, max=1) 
  
  if (B+C < 0.5) {
    p[i] = 1  
  }
  else {
    p[i] = 0
  }
}

paste("The probablity of B+C < 1/2: " , sum(p)/500)
## [1] "The probablity of B+C < 1/2:  0.12"

(b) BC < 1/2.

#  BC < 1/2.

p = vector("numeric", 500)
for (i in 1:500) {
  B = runif(1, min=0, max=1) 
  
  C = runif(1, min=0, max=1) 
  
  if (B*C < 0.5) {
    p[i] = 1  
  }
  else {
    p[i] = 0
  }
}

paste("The probablity of B*C < 1/2: " , sum(p)/500)
## [1] "The probablity of B*C < 1/2:  0.828"

(c) |B − C| < 1/2

#  |B-C| < 1/2.

p = vector("numeric", 500)
for (i in 1:500) {
  B = runif(1, min=0, max=1) 
  
  C = runif(1, min=0, max=1) 
  
  if (abs(B-C) < 0.5) {
    p[i] = 1  
  }
  else {
    p[i] = 0
  }
}

paste("The probablity of |B-C| < 1/2: " , sum(p)/500)
## [1] "The probablity of |B-C| < 1/2:  0.766"

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

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

p = vector("numeric", 500)
for (i in 1:500) {
  B = runif(1, min=0, max=1) 
  
  C = runif(1, min=0, max=1) 
  
  if (max(B, C) < 0.5) {
    p[i] = 1  
  }
  else {
    p[i] = 0
  }
}

paste("The probablity of max{B,C} < 1/2: " , sum(p)/500)
## [1] "The probablity of max{B,C} < 1/2:  0.244"

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

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

p = vector("numeric", 500)
for (i in 1:500) {
  B = runif(1, min=0, max=1) 
  
  C = runif(1, min=0, max=1) 
  
  if (min(B, C) < 0.5) {
    p[i] = 1  
  }
  else {
    p[i] = 0
  }
}

paste("The probablity of min{B,C} < 1/2: " , sum(p)/500)
## [1] "The probablity of min{B,C} < 1/2:  0.752"