A wooden stick was broken randomly into three pieces. What is the probability that a triangle can be built from those three parts?


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Question Number 12506 by Joel577 last updated on 24/Apr/17

$A wooden stick was broken randomly into$ $three pieces . What is the probability that a triangle$ $can be built from those three parts ?$
	Answered by mrW1 last updated on 25/Apr/17

	${L e t}^{'} s s a y t h e w o o d e n s t i c k i s b r o c k e n$ $i n t o t h r e e p i e c e s A, B, C .$ $T h e l e n g t h o f w o o d e n s t i c k i s 1 .$ $W h e n a w o o d e n s t i c k i s b r o c k e n i n t o$ $3 p i e c e s, t h e r e a r e 4 p o s s i b i l i t i e s :$ $1) n o n e o f t h e t h r e e p i e c e s i s l o n g e r$ $t h a n o r e q u a l t o 1 / 2;$ $2.1) o n e o f t h e t h r e e p i e c e s i s l o n g e r$ $t h a n o r e q u a l t o 1 / 2, a n d t h i s p i e c e i s A;$ $2.2) o n e o f t h e t h r e e p i e c e s i s l o n g e r$ $t h a n o r e q u a l t o 1 / 2, a n d t h i s p i e c e i s B;$ $2.3) o n e o f t h e t h r e e p i e c e s i s l o n g e r$ $t h a n o r e q u a l t o 1 / 2, a n d t h i s p i e c e i s C .$ $O n l y i n c a s e 1) t h e t h r e e p i e c e s c a n$ $b u i l d a t r i a n g l e .$ $\Rightarrow p r o b a b i l i t y = 1 / 4 .$
	Answered by ajfour last updated on 24/Apr/17

	$∙ b r e a k t h e s t i c k a n y w h e r e$ $∙ f o r s u c c e s s b r e a k t h e l a r g e r$ $o f t h e t w o p i e c e s$ $s o p r o b a b i l i t y t h a t a t r i a n g l e$ $i s f o r m e d = 1 \times (1 / 2) .$
	Commented by mrW1 last updated on 25/Apr/17

	$w h a t i f t h e s t i c k i s b r o k e n i n t o t w o$ $e q u a l p i e c e s ?$ $b r e a k t h e l a r g e r o f t h e t w o p i e c e s d o e s n t$ $a l w a y s m e a n s u c c e s s, e . g .$ $b r e a k 1 i n t o : 1 / 5, 4 / 5$ $b r e a k 4 / 5 i n t o : 1 / 5, 3 / 5$ $1 / 5, 1 / 5 a n d 3 / 5 c a n n o t b u i l d a t r i a n g l e .$
	Commented by ajfour last updated on 25/Apr/17

	$t h a n k s f o r t h e l i g h t .$
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