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Question Number 55539 by mr W last updated on 26/Feb/19

Commented by MJS last updated on 27/Feb/19

$what am I missing ? from any point A we have$ $\pm 45 ° range = 90 ° for B \Rightarrow the probability should$ $be 25 %$
Commented by mr W last updated on 26/Feb/19

$T w o p o i n t s A a n d B a r e s e l e c t e d$ $r a n d o m l y o n a c i r c l e w i t h c e n t e r O .$ $W h a t i s t h e p r o b a b i l i t y t h a t t h e$ $a n g l e ∠ A O B ⩽ 45 ° ?$
	Answered by kaivan.ahmadi last updated on 26/Feb/19

	$p = \frac{l (A B)}{2 π r} = \frac{\frac{2 π r}{8}}{2 π r} = \frac{1}{8}$
	Answered by mr W last updated on 26/Feb/19

	Commented by MJS last updated on 27/Feb/19

	$correct me if I^{'} m wrong, but your square$ ${doesn}^{'} t include pairs like θ_{1} = 330 °; θ_{2} = 15 °$ $I think you must add \frac{45 \times 45}{360} and then you$ $get \frac{1}{4} instead of \frac{15}{64}$
	Commented by mr W last updated on 26/Feb/19

	$t h e p o s i t i o n o f A c a n b e r e p r e s e n t e d$ $b y θ_{1} w i t h θ_{1} u n i f o r m l y d i s t r i b u t e d$ $b e t w e e n 0 a n d 360 ° .$ $t h e p o s i t i o n o f B c a n b e r e p r e s e n t e d$ $b y θ_{2} w i t h θ_{2} u n i f o r m l y d i s t r i b u t e d$ $b e t w e e n 0 a n d 360 ° .$ $φ = θ_{2} - θ_{1}$ $- 45 ° ⩽ φ ⩽ 45 °$ $\Rightarrow - 45 ° ⩽ θ_{2} - θ_{1} ⩽ 45 °$ $\Rightarrow θ_{2} - θ_{1} ⩾ - 45 °$ $\Rightarrow θ_{2} - θ_{1} ⩽ 45 °$
	Commented by mr W last updated on 26/Feb/19

	Commented by mr W last updated on 26/Feb/19

	$e a c h p o i n t P i n t h e s q u a r e 360 \times 360$ $r e p r e s e n t s a p o i n t p a i r A a n d B .$ $t h e a r e a o f s q u a r e 360 \times 360 r e p r e s e n t s$ $a l l p o s s i b i l i t i e s .$ $l i n e L_{1} : θ_{2} - θ_{1} = - 45 °$ $l i n e L_{2} : θ_{2} - θ_{1} = 45 °$ $i t m e a n s e a c h p o i n t i n t h e s h a d e d$ $a r e a f u l f i l l s t h e c o n d i t i o n$ $- 45 ° ⩽ θ_{2} - θ_{1} ⩽ 45 °$ $t h e r e f o r e t h e p r o b a b i l i t y f o r ∣ φ ∣⩽ 45 °$ $i s p = \frac{a r e a s h a d e d}{a r e a o f s q u a r e} = \frac{360 \times 360 - 315 \times 315}{360 \times 360}$ $w h i c h i s \frac{15}{64} \approx 0.234 .$
	Commented by kaivan.ahmadi last updated on 26/Feb/19

	$i t i s e q u v a l e n t t o :$ $i f w e h a v e a l i n e s e g m e n t w i t h l e n g t h 2 π r$ $a n d w e c h o s e t o w p o i n A, B$ $w h a t i s t h e p r o b a b l i t y t h a t A \bar{B} ⩽ \frac{π r}{4}$
	Commented by mr W last updated on 26/Feb/19
	I tried to simulate this question with the app Grapher and got following result: Among 10000 random selections 2339 fulfill the condiction, the probability is 0.2339. This is what I input: sum(if(abs(rnd360-rnd360)<=45,1,0),i=1..10000)
	Commented by mr W last updated on 26/Feb/19

	Commented by mr W last updated on 26/Feb/19

	$t o a h m a d i s i r :$ $y o u r i n t e r p r e t a t i o n i s c o r r e c t .$
	Commented by kaivan.ahmadi last updated on 26/Feb/19

	$i s i t t h e m e a n i n g o f p r o b a b l i t y ?$
	Commented by mr W last updated on 26/Feb/19

	$y e s .$
	Commented by mr W last updated on 27/Feb/19

	$y o u a r e r i g h t s i r . t h a n k s a l o t!$
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