Take a stick of unit length and break it into three pieces, choosing the break points at random. (The break points are assumed to be chosen simultane- ously.) What is the probability that the three pieces can be used to form a triangle? Hint: The sum of the lengths of any two pieces must exceed the length of the third, so each piece must have length < 1/2. Now use Exer-
Take a stick of unit length and break it into three pieces, choosing the break points at random. (The break points are assumed to be chosen simultane- ously.) What is the probability that the three pieces can be used to form a triangle? Hint: The sum of the lengths of any two pieces must exceed the length of the third, so each piece must have length < 1/2. Now use Exer-
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.3: Rules For Addition
Problem 8P
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Please do Q12 and please show work and show steps please
Transcribed Image Text:12 Take a stick of unit length and break it into three pieces, choosing the break
points at random. (The break points are assumed to be chosen simultane-
ously.) What is the probability that the three pieces can be used to form a
triangle? Hint: The sum of the lengths of any two pieces must exceed the
length of the third, so each piece must have length < 1/2. Now use Exer-
cise 8(g).
![8 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) B² + C² ≤ 1/2.
(i) (B − 1/2)² + (C − 1/2)² < 1/4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F5d6e96c4-018b-4e3f-93ef-76fe627f688a%2Fchrjkpa_processed.png&w=3840&q=75)
Transcribed Image Text:8 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) B² + C² ≤ 1/2.
(i) (B − 1/2)² + (C − 1/2)² < 1/4.
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