Introduction to Probability · Problem 2.2.8

Combinatorics — Grinstead & Snell

Mathematics — Probability · continuous-density-functions

Problem

Choose independently two numbers

B

and

C

at random from the interval

[0, 1]

with uniform density. Note that the point

(B, C)

is then chosen at random in the unit square. Find the probability that (a)

B + C < 1/2

. (b)

BC < 1/2

. (c)

|B - C| < 1/2

. (d)

\max\{B, C\} < 1/2

. (e)

\min\{B, C\} < 1/2

. (f)

B < 1/2

and

1 - C < 1/2

. (g) conditions (c) and (f) both hold. (h)

B^2 + C^2 \leq 1/2

. (i)

(B - 1/2)^2 + (C - 1/2)^2 < 1/4