diff --git a/is/UB1/ISUB1.tex b/is/UB1/ISUB1.tex index 1f51ce3..f473add 100644 --- a/is/UB1/ISUB1.tex +++ b/is/UB1/ISUB1.tex @@ -89,17 +89,17 @@ \item hands with exaclty two pairs: ${13\choose 2}{4\choose 2}^2\cdot 11\cdot 4=123552$\\ $\mathds{P}(2\ pairs)=\frac{123552}{2598960}=\frac{198}{4165}\approx 0.0475$ \item \begin{enumerate}[(a)] - \item hands with exaclty two pairs without spades: ${13\choose 2}{3\choose 2}^2\cdot 11\cdot 3=23166$\\ - possible hands ${39 \choose 5} = 575757$\\ - $\mathds{P}(2\ pairs|\neg Spades)=\frac{23166}{575757}=\frac{198}{4921}\approx 0.04023$\\ + \item possible hands without Spades: ${39\choose 5}=575757$\\ + hands with exaclty two pairs without Spades: ${13\choose 2}{3\choose 2}^2\cdot 11\cdot 3=23166$\\ + $\mathds{P}(2\ pairs|\neg Spades)=\frac{23166}{575757}=\frac{168}{4921}\approx 0.0402$\\ one pair:\\ ${13\choose 1}{3\choose 2}{12 \choose 3}\cdot 3^3=231660$\\ - $\mathds{P}(1\ pair|\neg Spades)=\frac{231660}{575757}=\frac{1980}{4921}\approx 0.40235$ + $\mathds{P}(1\ pair|\neg Spades)=\frac{231660}{575757}=\frac{1980}{4921}\approx 0.402$ \item $A:=2\ pairs,\ B:=\neg Spades$\\ $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$\\ $P(A)=\frac{198}{4165},\ P(B)=\frac{39}{52}\cdot\frac{38}{51}\cdot\frac{37}{50}\cdot\frac{36}{49}\cdot\frac{35}{48}=\frac{2109}{9520}$\\ $P(B|A)=1-P(\neg B| A)=1-\left(\frac{1}{4}+\frac{3}{4}\frac{1}{4}+\left(\frac{3}{4}\right)^2\frac{1}{4}+\left(\frac{3}{4}\right)^3\frac{1}{4}+\left(\frac{3}{4}\right)^4\frac{1}{3}\right)=1-\frac{101}{128}=\frac{27}{128}\approx 0.211$\\ - $P(A|B)=\frac{\frac{27}{128}\cdot \frac{198}{4165}}{\frac{2109}{9520}}$%TODO + $P(A|B)=\frac{\frac{27}{128}\cdot \frac{198}{4165}}{\frac{2109}{9520}}=\frac{891}{19684}\approx 0.0452$%TODO \end{enumerate} \end{enumerate} \Aufgabe{Random Variables}{2+2+2=6}