Grinstead-Snell-Probability · Problem 8

Combinatorics

Mathematics · 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