Условная вероятность. Формулы полной вероятности и Байеса

```7. &Oacute;&ntilde;&euml;&icirc;&acirc;&iacute;&agrave;&yuml; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;. &Ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave; &Aacute;&agrave;&eacute;&aring;&ntilde;&agrave;
7.1. &Iuml;&eth;&aring;&auml;&acirc;&agrave;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&aring; &ntilde;&acirc;&aring;&auml;&aring;&iacute;&egrave;&yuml;
&Iuml;&oacute;&ntilde;&ograve;&uuml; A &egrave; B &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&aring; &ntilde;&icirc;&aacute;&ucirc;&ograve;&egrave;&yuml; &egrave; P(B) &gt; 0, &ograve;&icirc;&atilde;&auml;&agrave; P(A|B) = P(AB)
P(B) &oacute;&ntilde;&euml;&icirc;&acirc;&iacute;&agrave;&yuml; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; A &iuml;&eth;&egrave; &oacute;&ntilde;&euml;&icirc;&acirc;&egrave;&egrave; B .
&Iuml;&oacute;&ntilde;&ograve;&uuml; A &iacute;&aring;&ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&aring; &ntilde;&icirc;&aacute;&ucirc;&ograve;&egrave;&aring;, H1 , . . . , Hn &iuml;&icirc;&iuml;&agrave;&eth;&iacute;&icirc; &iacute;&aring;&ntilde;&icirc;&acirc;&igrave;&aring;&ntilde;&ograve;&iacute;&ucirc;&aring;
&ntilde;&icirc;n
&aacute;&ucirc;&ograve;&egrave;&yuml; &ntilde; &iuml;&icirc;&euml;&icirc;&aelig;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&ucirc;&igrave;&egrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave; &egrave; &ograve;&agrave;&ecirc;&egrave;&aring;, &divide;&ograve;&icirc; A ⊂ j=1
∪ Hj . &Ograve;&icirc;&atilde;&auml;&agrave;
P(A) =
n
X
P(Hj )P(A|Hj )
(&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;)
j=1
&Icirc;&aacute;&ucirc;&divide;&iacute;&icirc; &ntilde;&icirc;&aacute;&ucirc;&ograve;&egrave;&yuml; H1 , . . . , Hnn &iacute;&agrave;&ccedil;&ucirc;&acirc;&agrave;&thorn;&ograve; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&ccedil;&agrave;&igrave;&egrave;. &times;&agrave;&ntilde;&ograve;&icirc; &eth;&agrave;&ntilde;&ntilde;&igrave;&agrave;&ograve;&eth;&egrave;&acirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &ograve;&agrave;&ecirc;&egrave;&aring; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&ccedil;&ucirc;, &divide;&ograve;&icirc; j=1
∪ Hj = Ω. &Acirc; &yacute;&ograve;&icirc;&igrave; &ntilde;&euml;&oacute;&divide;&agrave;&aring; &atilde;&icirc;&acirc;&icirc;&eth;&yuml;&ograve;, &divide;&ograve;&icirc; &icirc;&iacute;&egrave;
&icirc;&aacute;&eth;&agrave;&ccedil;&oacute;&thorn;&ograve; &iuml;&icirc;&euml;&iacute;&oacute;&thorn; &atilde;&eth;&oacute;&iuml;&iuml;&oacute; &ntilde;&icirc;&aacute;&ucirc;&ograve;&egrave;&eacute;.
&Oacute;&ntilde;&euml;&icirc;&acirc;&iacute;&agrave;&yuml; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; P(Hi |A) &eth;&agrave;&acirc;&iacute;&agrave;
P(Hi |A) =
P(AHi )
P(Hi )P(A|Hi )
= P
n
P(A)
P(Hj )P(A|Hj )
(&ocirc;&icirc;&eth;&igrave;&oacute;&euml;&agrave; &Aacute;&agrave;&eacute;&aring;&ntilde;&agrave;)
j=1
&Acirc; &ccedil;&agrave;&auml;&agrave;&divide;&agrave;&otilde; &iacute;&agrave; &ocirc;&icirc;&eth;&igrave;&oacute;&euml;&ucirc; &iuml;&icirc;&euml;&iacute;&icirc;&eacute; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &egrave; &Aacute;&agrave;&eacute;&aring;&ntilde;&agrave; &oacute;&ntilde;&euml;&icirc;&acirc;&iacute;&ucirc;&aring; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;
P(A|Hi ) &euml;&egrave;&aacute;&icirc; &ccedil;&agrave;&auml;&agrave;&iacute;&ucirc; &acirc; &ograve;&aring;&ecirc;&ntilde;&ograve;&aring;, &euml;&egrave;&aacute;&icirc; &euml;&aring;&atilde;&ecirc;&icirc; &acirc;&ucirc;&divide;&egrave;&ntilde;&euml;&yuml;&thorn;&ograve;&ntilde;&yuml;.
7.2. &Iuml;&eth;&agrave;&ecirc;&ograve;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &ccedil;&agrave;&iacute;&yuml;&ograve;&egrave;&aring;
&Acirc; &icirc;&auml;&iacute;&icirc;&igrave; &igrave;&agrave;&euml;&aring;&iacute;&uuml;&ecirc;&icirc;&igrave; &atilde;&icirc;&eth;&icirc;&auml;&ecirc;&aring; &iuml;&icirc;&euml;&egrave;&ouml;&egrave;&yuml; &eth;&agrave;&ccedil;&ucirc;&ntilde;&ecirc;&egrave;&acirc;&agrave;&aring;&ograve; &aacute;&eth;&icirc;&auml;&yuml;&atilde;&oacute;. &Igrave;&icirc;&aelig;&iacute;&icirc; &ntilde;&divide;&egrave;&ograve;&agrave;&ograve;&uuml;, &divide;&ograve;&icirc; &aring;&ntilde;&ograve;&uuml; &divide;&aring;&ograve;&ucirc;&eth;&aring; &oslash;&agrave;&iacute;&ntilde;&agrave; &egrave;&ccedil; &iuml;&yuml;&ograve;&egrave;, &divide;&ograve;&icirc; &icirc;&iacute; &iacute;&agrave;&otilde;&icirc;&auml;&egrave;&ograve;&ntilde;&yuml; &acirc; &icirc;&auml;&iacute;&icirc;&igrave; &egrave;&ccedil; &acirc;&icirc;&ntilde;&uuml;&igrave;&egrave;
&aacute;&agrave;&eth;&icirc;&acirc; &atilde;&icirc;&eth;&icirc;&auml;&ecirc;&agrave;, &aacute;&aring;&ccedil;&eth;&agrave;&ccedil;&euml;&egrave;&divide;&iacute;&icirc; &acirc; &ecirc;&agrave;&ecirc;&icirc;&igrave; &icirc;&iacute; &iacute;&aring; &icirc;&ograve;&auml;&agrave;&aring;&ograve; &iuml;&eth;&aring;&auml;&iuml;&icirc;&divide;&ograve;&aring;&iacute;&egrave;&yuml; &iacute;&egrave; &icirc;&auml;&iacute;&icirc;&igrave;&oacute;
&egrave;&ccedil; &iacute;&egrave;&otilde;. &Auml;&acirc;&icirc;&aring; &iuml;&icirc;&euml;&egrave;&ouml;&aring;&eacute;&ntilde;&ecirc;&egrave;&otilde; &iuml;&icirc;&ntilde;&aring;&ograve;&egrave;&euml;&egrave; &ntilde;&aring;&igrave;&uuml; &aacute;&agrave;&eth;&icirc;&acirc;, &iacute;&icirc; &aacute;&eth;&icirc;&auml;&yuml;&atilde;&oacute; &iacute;&aring; &icirc;&aacute;&iacute;&agrave;&eth;&oacute;&aelig;&egrave;&euml;&egrave;.
&Ecirc;&agrave;&ecirc;&icirc;&acirc;&ucirc; &oslash;&agrave;&iacute;&ntilde;&ucirc; &iacute;&agrave;&eacute;&ograve;&egrave; &aring;&atilde;&icirc; &acirc; &acirc;&icirc;&ntilde;&uuml;&igrave;&icirc;&igrave; &aacute;&agrave;&eth;&aring;?
&Yacute;&ograve;&icirc; &ccedil;&agrave;&auml;&agrave;&divide;&agrave; &iacute;&agrave; &icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&aring;&iacute;&egrave;&aring; &oacute;&ntilde;&euml;&icirc;&acirc;&iacute;&icirc;&eacute; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;.
1/3.
&Atilde;&eth;&oacute;&iuml;&iuml;&agrave; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc;, &ntilde;&auml;&agrave;&thorn;&ugrave;&agrave;&yuml; &yacute;&ecirc;&ccedil;&agrave;&igrave;&aring;&iacute;, &ntilde;&icirc;&ntilde;&ograve;&icirc;&egrave;&ograve; &egrave;&ccedil; 5 &icirc;&ograve;&euml;&egrave;&divide;&iacute;&egrave;&ecirc;&icirc;&acirc;, 10 &otilde;&icirc;&eth;&icirc;&oslash;&egrave;&otilde; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc; &egrave; 15 &ntilde;&euml;&agrave;&aacute;&ucirc;&otilde; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve;&icirc;&acirc;. &Icirc;&ograve;&euml;&egrave;&divide;&iacute;&egrave;&ecirc; &acirc;&ntilde;&aring;&atilde;&auml;&agrave; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve; &icirc;&ouml;&aring;&iacute;&ecirc;&oacute;
&frac34;&icirc;&ograve;&euml;&egrave;&divide;&iacute;&icirc;&iquest;, &otilde;&icirc;&eth;&icirc;&oslash;&egrave;&eacute; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve; &frac34;&icirc;&ograve;&euml;&egrave;&divide;&iacute;&icirc;&iquest; &egrave; &frac34;&otilde;&icirc;&eth;&icirc;&oslash;&icirc;&iquest; &ntilde; &eth;&agrave;&acirc;&iacute;&ucirc;&igrave;&egrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave;, &ntilde;&euml;&agrave;&aacute;&ucirc;&eacute; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve; &frac34;&otilde;&icirc;&eth;&icirc;&oslash;&icirc;&iquest;, &frac34;&oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&iquest; &egrave; &frac34;&iacute;&aring;&oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&iquest; &ntilde; &eth;&agrave;&acirc;&iacute;&ucirc;&igrave;&egrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&yuml;&igrave;&egrave;. &Icirc;&iuml;&eth;&aring;&auml;&aring;&euml;&egrave;&ograve;&uuml; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc;
&iacute;&agrave;&oacute;&atilde;&agrave;&auml; &acirc;&ucirc;&ccedil;&acirc;&agrave;&iacute;&iacute;&ucirc;&eacute; &ntilde;&ograve;&oacute;&auml;&aring;&iacute;&ograve; &iuml;&icirc;&euml;&oacute;&divide;&egrave;&ograve; &icirc;&ouml;&aring;&iacute;&ecirc;&oacute; &frac34;&icirc;&ograve;&euml;&egrave;&divide;&iacute;&icirc;&iquest;, &frac34;&otilde;&icirc;&eth;&icirc;&oslash;&icirc;&iquest;, &frac34;&oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&iquest;, &frac34;&iacute;&aring;&oacute;&auml;&icirc;&acirc;&euml;&aring;&ograve;&acirc;&icirc;&eth;&egrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc;&iquest;.
1/3, 1/3, 1/6, 1/6.
&times;&egrave;&ntilde;&euml;&icirc; &atilde;&eth;&oacute;&ccedil;&icirc;&acirc;&ucirc;&otilde; &igrave;&agrave;&oslash;&egrave;&iacute;, &iuml;&eth;&icirc;&aring;&ccedil;&aelig;&agrave;&thorn;&ugrave;&egrave;&otilde; &igrave;&egrave;&igrave;&icirc; &aacute;&aring;&iacute;&ccedil;&icirc;&ecirc;&icirc;&euml;&icirc;&iacute;&ecirc;&egrave;, &icirc;&ograve;&iacute;&icirc;&ntilde;&egrave;&ograve;&ntilde;&yuml;
&ecirc; &divide;&egrave;&ntilde;&euml;&oacute; &euml;&aring;&atilde;&ecirc;&icirc;&acirc;&ucirc;&otilde; &igrave;&agrave;&oslash;&egrave;&iacute; &ecirc;&agrave;&ecirc; 3:2. &Acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &atilde;&eth;&oacute;&ccedil;&icirc;&acirc;&agrave;&yuml; &igrave;&agrave;&oslash;&egrave;&iacute;&agrave;
&aacute;&oacute;&auml;&aring;&ograve; &ccedil;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&yuml;&ograve;&uuml;&ntilde;&yuml;, &eth;&agrave;&acirc;&iacute;&agrave; 0.1, &agrave; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &aacute;&oacute;&auml;&aring;&ograve; &ccedil;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&yuml;&ograve;&uuml;&ntilde;&yuml; &euml;&aring;&atilde;&ecirc;&icirc;&acirc;&agrave;&yuml; 1.
&Oacute;&ecirc;&agrave;&ccedil;&agrave;&iacute;&egrave;&aring;.
&Icirc;&ograve;&acirc;&aring;&ograve;.
2.
&Icirc;&ograve;&acirc;&aring;&ograve;.
3.
1
0.2. &Oacute; &aacute;&aring;&iacute;&ccedil;&icirc;&ecirc;&icirc;&euml;&icirc;&iacute;&ecirc;&egrave; &ccedil;&agrave;&iuml;&eth;&agrave;&acirc;&euml;&yuml;&aring;&ograve;&ntilde;&yuml; &igrave;&agrave;&oslash;&egrave;&iacute;&agrave;. &Iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &yacute;&ograve;&icirc;
&atilde;&eth;&oacute;&ccedil;&icirc;&acirc;&agrave;&yuml; &igrave;&agrave;&oslash;&egrave;&iacute;&agrave;.
3/7.
&Auml;&acirc;&agrave; &icirc;&otilde;&icirc;&ograve;&iacute;&egrave;&ecirc;&agrave; &icirc;&auml;&iacute;&icirc;&acirc;&eth;&aring;&igrave;&aring;&iacute;&iacute;&icirc; &egrave; &iacute;&aring;&ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&igrave;&icirc; &ntilde;&ograve;&eth;&aring;&euml;&yuml;&thorn;&ograve; &acirc; &ecirc;&agrave;&aacute;&agrave;&iacute;&agrave;. &Egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&iacute;&icirc;,
&divide;&ograve;&icirc; &iuml;&aring;&eth;&acirc;&ucirc;&eacute; &iuml;&icirc;&iuml;&agrave;&auml;&agrave;&aring;&ograve; &ntilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.8, &agrave; &acirc;&ograve;&icirc;&eth;&icirc;&eacute; &ntilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.4. &Ecirc;&agrave;&aacute;&agrave;&iacute;
&oacute;&aacute;&egrave;&ograve;, &egrave; &acirc; &iacute;&frac14;&igrave; &icirc;&aacute;&iacute;&agrave;&eth;&oacute;&aelig;&aring;&iacute;&agrave; &icirc;&auml;&iacute;&agrave; &iuml;&oacute;&euml;&yuml;. &Ecirc;&agrave;&ecirc; &auml;&aring;&euml;&egrave;&ograve;&uuml; &ecirc;&agrave;&aacute;&agrave;&iacute;&agrave;?
&Acirc; &iuml;&eth;&icirc;&iuml;&icirc;&eth;&ouml;&egrave;&egrave; 6:1.
&Acirc; &iuml;&oacute;&iacute;&ecirc;&ograve;&aring; &iuml;&eth;&icirc;&ecirc;&agrave;&ograve;&agrave; &egrave;&igrave;&aring;&aring;&ograve;&ntilde;&yuml; 10 &ograve;&aring;&euml;&aring;&acirc;&egrave;&ccedil;&icirc;&eth;&icirc;&acirc;, &auml;&euml;&yuml; &ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &egrave;&ntilde;&iuml;&eth;&agrave;&acirc;&iacute;&icirc;&eacute; &eth;&agrave;&aacute;&icirc;&ograve;&ucirc; &acirc; &ograve;&aring;&divide;&aring;&iacute;&egrave;&aring; &igrave;&aring;&ntilde;&yuml;&ouml;&agrave; &eth;&agrave;&acirc;&iacute;&agrave; 0.9, &egrave; 5 &ograve;&aring;&euml;&aring;&acirc;&egrave;&ccedil;&icirc;&eth;&icirc;&acirc; &ntilde; &agrave;&iacute;&agrave;&euml;&icirc;&atilde;&egrave;&divide;&iacute;&icirc;&eacute;
&acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.95. &Iacute;&agrave;&eacute;&ograve;&egrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &auml;&acirc;&agrave; &ograve;&aring;&euml;&aring;&acirc;&egrave;&ccedil;&icirc;&eth;&agrave;, &acirc;&ccedil;&yuml;&ograve;&ucirc;&aring; &iacute;&agrave;&oacute;&auml;&agrave;&divide;&oacute; &acirc; &iuml;&oacute;&iacute;&ecirc;&ograve;&aring; &iuml;&eth;&icirc;&ecirc;&agrave;&ograve;&agrave;, &aacute;&oacute;&auml;&oacute;&ograve; &eth;&agrave;&aacute;&icirc;&ograve;&agrave;&ograve;&uuml; &egrave;&ntilde;&iuml;&eth;&agrave;&acirc;&iacute;&icirc; &acirc; &ograve;&aring;&divide;&aring;&iacute;&egrave;&aring; &igrave;&aring;&ntilde;&yuml;&ouml;&agrave;.
3529
4200 ≈ 0.8402.
&Icirc;&ograve;&acirc;&aring;&ograve;.
4.
&Icirc;&ograve;&acirc;&aring;&ograve;.
5.
&Icirc;&ograve;&acirc;&aring;&ograve;.
7.3. &Auml;&icirc;&igrave;&agrave;&oslash;&iacute;&aring;&aring; &ccedil;&agrave;&auml;&agrave;&iacute;&egrave;&aring;
&Acirc; &atilde;&icirc;&eth;&icirc;&auml;&aring; N &auml;&icirc;&aelig;&auml;&euml;&egrave;&acirc;&ucirc;&igrave;&egrave; &aacute;&ucirc;&acirc;&agrave;&thorn;&ograve; &divide;&aring;&ograve;&acirc;&aring;&eth;&ograve;&uuml; &acirc;&ntilde;&aring;&otilde; &auml;&iacute;&aring;&eacute;. &Ecirc;&eth;&icirc;&igrave;&aring; &ograve;&icirc;&atilde;&icirc;, &ccedil;&agrave;&igrave;&aring;&divide;&aring;&iacute;&icirc;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; &acirc; &ecirc;&agrave;&ecirc;&icirc;&eacute;-&ograve;&icirc; &auml;&aring;&iacute;&uuml; &auml;&icirc;&aelig;&auml;&uuml; &oslash;&aring;&euml;, &ograve;&icirc; &acirc; &auml;&acirc;&oacute;&otilde; &ntilde;&euml;&oacute;&divide;&agrave;&yuml;&otilde; &egrave;&ccedil; &ograve;&eth;&aring;&otilde; &icirc;&iacute;
&aacute;&oacute;&auml;&aring;&ograve; &egrave;&auml;&ograve;&egrave; &egrave; &iacute;&agrave; &ntilde;&euml;&aring;&auml;&oacute;&thorn;&ugrave;&egrave;&eacute; &auml;&aring;&iacute;&uuml;. &times;&aring;&igrave;&oacute; &eth;&agrave;&acirc;&iacute;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &ccedil;&agrave;&acirc;&ograve;&eth;&agrave;
&aacute;&oacute;&auml;&aring;&ograve; &otilde;&icirc;&eth;&icirc;&oslash;&agrave;&yuml; &iuml;&icirc;&atilde;&icirc;&auml;&agrave;, &aring;&ntilde;&euml;&egrave; &ntilde;&aring;&atilde;&icirc;&auml;&iacute;&yuml; &auml;&icirc;&aelig;&auml;&yuml; &iacute;&aring; &aacute;&ucirc;&euml;&icirc;?
8/9.
&Iacute;&agrave; &ocirc;&agrave;&aacute;&eth;&egrave;&ecirc;&aring;, &egrave;&ccedil;&atilde;&icirc;&ograve;&icirc;&acirc;&euml;&yuml;&thorn;&ugrave;&aring;&eacute; &aacute;&icirc;&euml;&ograve;&ucirc;, &iuml;&aring;&eth;&acirc;&agrave;&yuml; &igrave;&agrave;&oslash;&egrave;&iacute;&agrave; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&auml;&egrave;&ograve; 25%,
&acirc;&ograve;&icirc;&eth;&agrave;&yuml; 35%, &ograve;&eth;&aring;&ograve;&uuml;&yuml; 40% &acirc;&ntilde;&aring;&otilde; &egrave;&ccedil;&auml;&aring;&euml;&egrave;&eacute;. &Acirc; &egrave;&ccedil; &iuml;&eth;&icirc;&auml;&oacute;&ecirc;&ouml;&egrave;&egrave; &aacute;&eth;&agrave;&ecirc; &ntilde;&icirc;&ntilde;&ograve;&agrave;&acirc;&euml;&yuml;&aring;&ograve;
5, 4 &egrave; 2%.
• &Ecirc;&agrave;&ecirc;&icirc;&acirc;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &ntilde;&euml;&oacute;&divide;&agrave;&eacute;&iacute;&icirc; &acirc;&ucirc;&aacute;&eth;&agrave;&iacute;&iacute;&ucirc;&eacute; &aacute;&icirc;&euml;&ograve; &auml;&aring;&ocirc;&aring;&ecirc;&ograve;&iacute;&ucirc;&eacute;?
• &Ntilde;&euml;&oacute;&divide;&agrave;&eacute;&iacute;&icirc; &acirc;&ucirc;&aacute;&eth;&agrave;&iacute;&iacute;&ucirc;&eacute; &aacute;&icirc;&euml;&ograve; &icirc;&ecirc;&agrave;&ccedil;&agrave;&euml;&ntilde;&yuml; &auml;&aring;&ocirc;&aring;&ecirc;&ograve;&iacute;&ucirc;&igrave;. &Ecirc;&agrave;&ecirc;&icirc;&acirc;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;
&ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &icirc;&iacute; &aacute;&ucirc;&euml; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&aring;&auml;&frac14;&iacute; &iuml;&aring;&eth;&acirc;&icirc;&eacute;, &acirc;&ograve;&icirc;&eth;&icirc;&eacute;, &ograve;&eth;&aring;&ograve;&uuml;&aring;&eacute; &igrave;&agrave;&oslash;&egrave;&iacute;&icirc;&eacute;?
140
80
0.0345; 125
345 , 345 , 345 .
&Acirc; &ecirc;&icirc;&eth;&icirc;&aacute;&ecirc;&aring; &euml;&aring;&aelig;&agrave;&ograve; 3 &iacute;&icirc;&acirc;&ucirc;&otilde; &egrave; 3 &oacute;&aelig;&aring; &egrave;&ntilde;&iuml;&icirc;&euml;&uuml;&ccedil;&icirc;&acirc;&agrave;&iacute;&iacute;&ucirc;&otilde; &ograve;&aring;&iacute;&iacute;&egrave;&ntilde;&iacute;&ucirc;&otilde; &igrave;&yuml;&divide;&agrave;. &Auml;&euml;&yuml;
&iuml;&aring;&eth;&acirc;&icirc;&eacute; &egrave;&atilde;&eth;&ucirc; &iacute;&agrave;&oacute;&auml;&agrave;&divide;&oacute; &aacute;&aring;&eth;&oacute;&ograve; &egrave;&ccedil; &ecirc;&icirc;&eth;&icirc;&aacute;&ecirc;&egrave; 2 &igrave;&yuml;&divide;&agrave; &egrave; &ccedil;&agrave;&ograve;&aring;&igrave; &egrave;&otilde; &acirc;&icirc;&ccedil;&acirc;&eth;&agrave;&ugrave;&agrave;&thorn;&ograve; &acirc;
&ecirc;&icirc;&eth;&icirc;&aacute;&ecirc;&oacute;. &Ecirc;&agrave;&ecirc;&icirc;&acirc;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &auml;&euml;&yuml; &acirc;&ograve;&icirc;&eth;&icirc;&eacute; &egrave;&atilde;&eth;&ucirc; &egrave;&ccedil; &yacute;&ograve;&icirc;&eacute; &ecirc;&icirc;&eth;&icirc;&aacute;&ecirc;&egrave; &iacute;&agrave;&oacute;&auml;&agrave;&divide;&oacute;
&acirc;&ucirc;&iacute;&oacute;&ograve;&uuml; 2 &iacute;&icirc;&acirc;&ucirc;&otilde; &igrave;&yuml;&divide;&agrave;?
2/25.
&Acirc; &oacute;&eth;&iacute;&aring; &euml;&aring;&aelig;&agrave;&euml;&icirc; 8 &aacute;&aring;&euml;&ucirc;&otilde; &egrave; 4 &divide;&frac14;&eth;&iacute;&ucirc;&otilde; &oslash;&agrave;&eth;&agrave;. &Icirc;&auml;&egrave;&iacute; &oslash;&agrave;&eth; &iuml;&icirc;&ograve;&aring;&eth;&yuml;&iacute;, &egrave; &ouml;&acirc;&aring;&ograve; &aring;&atilde;&icirc;
&iacute;&aring;&egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&aring;&iacute;. &Egrave;&ccedil; &oacute;&eth;&iacute;&ucirc; &aacute;&aring;&ccedil; &acirc;&icirc;&ccedil;&acirc;&eth;&agrave;&ugrave;&aring;&iacute;&egrave;&yuml; &egrave;&ccedil;&acirc;&euml;&aring;&ecirc;&euml;&egrave; &auml;&acirc;&agrave; &oslash;&agrave;&eth;&agrave;, &egrave; &icirc;&aacute;&agrave; &icirc;&ecirc;&agrave;&ccedil;&agrave;&euml;&egrave;&ntilde;&uuml;
&aacute;&aring;&euml;&ucirc;&igrave;&egrave;. &Ecirc;&agrave;&ecirc;&icirc;&acirc;&agrave; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml; &ograve;&icirc;&atilde;&icirc;, &divide;&ograve;&icirc; &aacute;&ucirc;&euml; &iuml;&icirc;&ograve;&aring;&eth;&yuml;&iacute; &divide;&frac14;&eth;&iacute;&ucirc;&eacute; &oslash;&agrave;&eth;? &Ecirc;&agrave;&ecirc;&icirc;&acirc;
&aacute;&oacute;&auml;&aring;&ograve; &icirc;&ograve;&acirc;&aring;&ograve; &ccedil;&agrave;&auml;&agrave;&divide;&egrave;, &aring;&ntilde;&euml;&egrave; &eth;&aring;&ccedil;&oacute;&euml;&uuml;&ograve;&agrave;&ograve; &egrave;&ccedil;&acirc;&euml;&aring;&divide;&aring;&iacute;&egrave;&yuml; &auml;&acirc;&oacute;&otilde; &oslash;&agrave;&eth;&icirc;&acirc; &iacute;&aring;&egrave;&ccedil;&acirc;&aring;&ntilde;&ograve;&aring;&iacute;?
2/5; 1/3.
&Ntilde;&ograve;&eth;&aring;&euml;&icirc;&ecirc; A &iuml;&icirc;&eth;&agrave;&aelig;&agrave;&aring;&ograve; &igrave;&egrave;&oslash;&aring;&iacute;&uuml; &ntilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.6, &ntilde;&ograve;&eth;&aring;&euml;&icirc;&ecirc; B &ntilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.5, &ntilde;&ograve;&eth;&aring;&euml;&icirc;&ecirc; C &ntilde; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&uuml;&thorn; 0.4. &Ntilde;&ograve;&eth;&aring;&euml;&ecirc;&egrave; &auml;&agrave;&euml;&egrave; &ccedil;&agrave;&euml;&iuml; &iuml;&icirc; &igrave;&egrave;&oslash;&aring;&iacute;&egrave;,
&egrave; &auml;&acirc;&aring; &iuml;&oacute;&euml;&egrave; &iuml;&icirc;&iuml;&agrave;&euml;&egrave; &acirc; &ouml;&aring;&euml;&uuml;. &times;&ograve;&icirc; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&aring;&aring;: &iuml;&icirc;&iuml;&agrave;&euml; &ntilde;&ograve;&eth;&aring;&euml;&icirc;&ecirc; C &acirc; &igrave;&egrave;&oslash;&aring;&iacute;&uuml; &egrave;&euml;&egrave;
&iacute;&aring;&ograve;? &Iacute;&agrave;&eacute;&ograve;&egrave; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&oacute;&thorn;&ugrave;&egrave;&aring; &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&icirc;&ntilde;&ograve;&egrave;.
10/19 &egrave; 9/19, &ograve;. &aring;. &acirc;&aring;&eth;&icirc;&yuml;&ograve;&iacute;&aring;&aring;, &divide;&ograve;&icirc; C &iuml;&icirc;&iuml;&agrave;&euml; &acirc; &igrave;&egrave;&oslash;&aring;&iacute;&uuml;.
6.
&Icirc;&ograve;&acirc;&aring;&ograve;.
7.
&Icirc;&ograve;&acirc;&aring;&ograve;.
8.
&Icirc;&ograve;&acirc;&aring;&ograve;.
9.
&Icirc;&ograve;&acirc;&aring;&ograve;.
10.
&Icirc;&ograve;&acirc;&aring;&ograve;.
2
```