7.1 Basic Probability Theory

$B3NN($N4pACM}O@$N2r@b(B

• Probability
  • Events($B;v>](B)$B$N$*$3$j$d$9$5(B, 0$B$+$i(B1$B$NCM(B
  • random variable
  • coin toss $B$NNc(B
• Conditional Probability
  • e' $B$,5/$-$?>r7o$N$b$H$G(Be $B$,5/$-$k3NN((B, $B>r7oIU$-3NN((B
    
    
    $$\begin{eqnarray*}PROB(e\vert e') = \frac{PROB(e\&e')}{PROB(e')} \end{eqnarray*}$$
    
    
  • horse race $B$NNc(B
    race $B?t(B 100, $B1+$N2s?t(B 30, $B1+$NF|$K>!$C$?2s?t(B15,$B1+$NF|$K>!$D3NN($O(B?
    
    
    $$\begin{eqnarray*}PROB(Win\vert Rain) &=& \frac{PROB(Win\&Rain)}{PROB(Rain)} \\ &=& 0.15/0.30 \\ &=& 0.5 \end{eqnarray*}$$
    
    
• Bayes's Rule
  • $PROB(A\vert B) \leftrightarrow PROB(B\vert A)$ $B$NAj8_JQ49(B
    
    
    $$\begin{eqnarray*}PROB(A\vert B) &=& \frac{PROB(A\&B)}{PROB(B)} \\ PROB(B\vert A) &=& \frac{PROB(A\&B)}{PROB(A)} \end{eqnarray*}$$
    
    $B$+$i(B $PROB(A\&B)$ $B$r>C5n(B
    
    $$\begin{eqnarray*}PROB(A\vert B) &=& \frac{PROB(B\vert A) \cdot PROB(A)}{PROB(B)} \end{eqnarray*}$$
    
    
  • horse race $B$NNc(B
    $B>!$C$?F|$,1+$G$"$k3NN($O(B?
    
    $$\begin{eqnarray*}PROB(Rain\vert Win) &=& \frac{PROB(Win\vert Rain) \cdot PROB(Rain)}{PROB(Win)} \\ &=& 0.5\cdot0.30/0.2 \\ &=& 0.75 \end{eqnarray*}$$
    
    $BD>@\$b$H$a$F$b(B,$B$b$A$m$s$*$J$87k2L(B
    
    $$\begin{eqnarray*}PROB(Rain\vert Win) &=& \frac{PROB(Rain\&Win)}{PROB(Win)} \\ \ &=& 0.15/0.2 \\ &=& 0.75 \end{eqnarray*}$$
    
    
  • $BI,MW$JE}7W(Bdata$B$,F@$i$l$J$$>l9g!"JL$NE}7W(Bdata$B$+$iJQ49$G$-$k(B
• Independence
  • $B;v>](B A, B $B$,FHN)$G$"$k(B $\leftrightarrow$ PROB(A|B) = P(A)
    $\leftrightarrow$ $PROB(A\&B) = PROB(A) \cdot PROB(B)$
  • horse race $B$NNc(B
    $B1+$,9_$k(B,$B>!$D(B $B$H$$$&;v>]$,$=$l$>$lFHN)$@$H$7$?$i(B,$B1+$NF|$K(B $B>!$D3NN($O(B,
    $0.2 \cdot 0.3 = 0.06$, $B$7$+$7]$O6/$$0MB84X78$K$"$k(B
    
• Application related to NL
  flies $B$,F0;l(B(V) $B$G;H$o$l$d$9$$$+(B, $BL>;l(B(N)$B$G;H$o$l$d$9$$$+(B?
  PROB(N|flies), PROB(V|flies) $B$r7W;;$9$l$P$h$$(B
  
  corpus $B$K$h$k$H(B.. 1,273,00 words, flies $B$NIQEY(B 1,000,
  $B$=$N$&$A(B V $B$K$J$C$F$k$b$N(B 600, N $B$K$J$C$F$k$b$N(B 400
  
  $$\begin{eqnarray*}PROB(N\vert flies) &=& \frac{PROB(N\&flies)}{PROB(flies)} \\ ... ...lies)} \\ &=& (600/1,273,000)/(1000/1,273,000) \\ &=& 0.625 \end{eqnarray*}$$
  
  $BHs>o$K(B poor $B$J