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[SPAM] [obm-l] Probabilidades Geométricas: 2 problemas difíceis

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1=BA Problema - este =E9 MUITO dif=EDcil!

=20

Considere uma caixa de base quadrada, cujos lados (da base) s=E3o =
unit=E1rios.
Na base desta caixa, s=E3o tra=E7ados dois segmentos de reta:

1)     A pr=F3pria diagonal da base; e

2)     O segmento de reta entre os pontos m=E9dios de dois lados =
opostos.

=20

Toma-se uma agulha de comprimento tamb=E9m unit=E1rio e joga-se, =
aleatoriamente,
dentro da caixa.

=20

Pergunta-se:

=20

Qual =E9 a probabilidade da agulha, ent=E3o pousada horizontalmente na =
base da
caixa (por hip=F3tese!), interceptar (em um ponto qualquer) o segmento =
de reta
de n=FAmero =931=94, descrito acima? E o de n=FAmero =932=94?

=20

Veja um problema an=E1logo (mas, mais f=E1cil!) em:

 <http://www.cut-the-knot.com/fta/Buffon/buffon9.html>
http://www.cut-the-knot.com/fta/Buffon/buffon9.html
=20
=20
2=BA Problema - este tamb=E9m =E9 dif=EDcil, mas n=E3o tanto quanto o =
primeiro.
=20
Considere um tri=E2ngulo eq=FCil=E1tero. Calcule a probabilidade de um =
segmento de
reta, determinado por um ponto qualquer de um dos lados desse =
tri=E2ngulo e
por outro ponto qualquer de um dos outros dois lados adjacentes, ser =
maior
do que a altura do tri=E2ngulo.

=20

Paradoxo de Bertrand (Bertrand's Paradox): =93Given a circle. Find the
probability that a chord chosen at random be longer than the side of an
inscribed equilateral triangle=94.

Refer=EAncia na Internet:  <http://www.cut-the-knot.com/bertrand.html>
http://www.cut-the-knot.com/bertrand.html


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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
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<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><SPAN =
class=3D281583313-28062008><FONT=20
face=3DVerdana size=3D2>1=BA Problema - este =E9 MUITO =
dif=EDcil!</FONT></SPAN></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana=20
size=3D2></FONT></SPAN>&nbsp;</P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana><FONT=20
size=3D2>Considere uma caixa de base quadrada, cujos lados (da base) =
s=E3o=20
unit=E1rios. Na base desta caixa, s=E3o tra=E7ados dois segmentos de=20
reta:<?xml:namespace prefix =3D o ns =3D =
"urn:schemas-microsoft-com:office:office"=20
/><o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 =
level1 lfo1; tab-stops: list 18.0pt"><FONT=20
face=3DVerdana><FONT size=3D2><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'; mso-fareast-font-family: =
'Lucida Sans Unicode'"><SPAN=20
style=3D"mso-list: Ignore">1)<SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN style=3D"FONT-FAMILY: 'Lucida Sans =
Unicode'">A pr=F3pria=20
diagonal da base; e<o:p></o:p></SPAN></FONT></FONT></P>
<P class=3DMsoNormal=20
style=3D"MARGIN: 0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; mso-list: l0 =
level1 lfo1; tab-stops: list 18.0pt"><FONT=20
face=3DVerdana><FONT size=3D2><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'; mso-fareast-font-family: =
'Lucida Sans Unicode'"><SPAN=20
style=3D"mso-list: Ignore">2)<SPAN=20
style=3D"FONT: 7pt 'Times New Roman'">&nbsp;&nbsp;&nbsp;&nbsp;=20
</SPAN></SPAN></SPAN><SPAN style=3D"FONT-FAMILY: 'Lucida Sans =
Unicode'">O segmento=20
de reta entre os pontos m=E9dios de dois lados=20
opostos.<o:p></o:p></SPAN></FONT></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><o:p><FONT face=3DVerdana=20
size=3D2>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana><FONT=20
size=3D2>Toma-se uma agulha de comprimento tamb=E9m unit=E1rio e =
joga-se,=20
aleatoriamente, dentro da caixa.<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><o:p><FONT face=3DVerdana=20
size=3D2>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana><FONT=20
size=3D2>Pergunta-se:<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><o:p><FONT face=3DVerdana=20
size=3D2>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana><FONT =
size=3D2>Qual=20
=E9 a probabilidade da agulha, ent=E3o pousada horizontalmente na base =
da caixa (por=20
hip=F3tese!), interceptar (em um ponto qualquer) o segmento de reta de =
n=FAmero=20
=93<?xml:namespace prefix =3D st1 ns =3D =
"urn:schemas-microsoft-com:office:smarttags"=20
/><st1:metricconverter w:st=3D"on" =
ProductID=3D"1=94">1=94</st1:metricconverter>,=20
descrito acima? E o de n=FAmero =93<st1:metricconverter w:st=3D"on"=20
ProductID=3D"2=94">2=94</st1:metricconverter>?<o:p></o:p></FONT></FONT></=
SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><o:p><FONT face=3DVerdana=20
size=3D2>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: 'Lucida Sans Unicode'"><FONT=20
face=3DVerdana>Veja um problema an=E1logo (mas, mais f=E1cil!)=20
em:<o:p></o:p></FONT></SPAN></P><SPAN=20
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: 'Lucida Sans Unicode'; =
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href=3D"http://www.cut-the-knot.com/fta/Buffon/buffon9.html";><FONT=20
face=3DVerdana>http://www.cut-the-knot.com/fta/Buffon/buffon9.html</FONT>=
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<DIV align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D281583313-28062008></SPAN></FONT>&nbsp;</DIV>
<DIV align=3Dleft><FONT face=3DVerdana size=3D2><SPAN =
class=3D281583313-28062008>2=BA=20
Problema - este tamb=E9m =E9 dif=EDcil, mas n=E3o tanto quanto o=20
primeiro.</SPAN></FONT></DIV>
<DIV align=3Dleft><FONT face=3DVerdana size=3D2><SPAN=20
class=3D281583313-28062008></SPAN></FONT>&nbsp;</DIV>
<DIV align=3Dleft><FONT><SPAN class=3D281583313-28062008>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><FONT face=3DVerdana><FONT=20
size=3D2>Considere um tri=E2ngulo eq=FCil=E1tero. Calcule a =
probabilidade de um segmento=20
de reta, determinado por um ponto qualquer de um dos lados desse =
tri=E2ngulo e por=20
outro ponto qualquer de um dos outros dois lados adjacentes, ser maior =
do que a=20
altura do tri=E2ngulo.<o:p></o:p></FONT></FONT></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-FAMILY: 'Lucida Sans Unicode'"><o:p><FONT face=3DVerdana=20
size=3D2>&nbsp;</FONT></o:p></SPAN></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><FONT =
face=3DVerdana><SPAN=20
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: 'Lucida Sans =
Unicode'">Paradoxo</SPAN><SPAN=20
lang=3DEN-US=20
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: 'Lucida Sans Unicode'; =
mso-ansi-language: EN-US">=20
de Bertrand (Bertrand's Paradox): =93Given a circle. Find the =
probability that a=20
chord chosen at random be longer than the side of an inscribed =
equilateral=20
triangle=94.<o:p></o:p></SPAN></FONT></P>
<P class=3DMsoNormal style=3D"MARGIN: 0cm 0cm 0pt"><SPAN=20
style=3D"FONT-SIZE: 10pt; FONT-FAMILY: 'Lucida Sans Unicode'"><FONT=20
face=3DVerdana>Refer=EAncia na Internet: </FONT><A=20
href=3D"http://www.cut-the-knot.com/bertrand.html";><FONT=20
face=3DVerdana>http://www.cut-the-knot.com/bertrand.html</FONT></A></SPAN=
></SPAN></FONT></P></DIV></BODY></HTML>

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