 |
$\begin{align}(i)&S=\{1 , 2 , 3 , 4 , 5 , 6 \} \\[5pt]
&A=\{ 2 , 3 , 5 \} \\[5pt]
&B=\{ 1 , 4 \} \\[5pt]
&C=\{5 , 6 \} \\[5pt]
&D=\{6 \} \\[5pt]
&\text {Pairs of mutually exclusive events are A and B ,} \\[5pt]
&\text {A and D , B and C , B and D} \\[5pt]
\end{align}$ | | |
 |
$\begin{align}&P(X)=\frac{1}{4} , P(Y)=\frac{5}{6} , P(X\cap Y)=\frac{1}{6} \\[5pt]
(i)P(X\cup Y)&=P(X)+P(Y)-P(X\cap Y) \\[5pt]
&=\frac {1}{4}+\frac {5}{6}+\frac {1}{6}\\[5pt]
&=\frac {3+10-2}{12}\\[5pt]
&=\underline{\underline{\frac{11}{12}}}
\end{align}$
| $\begin{align}(ii)P(X')&=1-P(X) \\[5pt]
&=1-\frac {1}{4}\\[5pt]
&=\underline{\underline{\frac{3}{4}}}
\end{align}$
| $\begin{align}(iii)P(Y')&=1-P(Y) \\[5pt]
&=1-\frac {5}{6}\\[5pt]
&=\underline{\underline{\frac{1}{6}}}
\end{align}$
|
$\begin{align}(iv)P[(X\cap Y)']&=1-P(X\cap Y) \\[5pt]
&=1-\frac {1}{6}\\[5pt]
&=\underline{\underline{\frac{5}{6}}}
\end{align}$
| $\begin{align}(v)P[(X\cup Y)']&=1-P(X\cup Y) \\[5pt]
&=1-\frac {11}{12}\\[5pt]
&=\underline{\underline{\frac{1}{12}}}
\end{align}$
| |
 |
$\begin{align}&P(A)=\frac{2}{7} , P(B')=\frac{1}{4} \\[5pt]
(i)P(A')&=1-P(A) \\[5pt]
&=1-\frac {2}{7}\\[5pt]
&=\underline{\underline{\frac{5}{7}}}
\end{align}$
| $\begin{align}(ii)P(B)&=1-P(B') \\[5pt]
&=1-\frac {1}{4}\\[5pt]
&=\underline{\underline{\frac{3}{4}}}
\end{align}$ | |
 |
$\begin{align}&P(X)=\frac{1}{2} , P(Y)=\frac{1}{3} , P(X\cup Y)=\frac{5}{6} \\[5pt]
(i)P(X\cup Y)&=P(X)+P(Y)-P(X\cap Y) \\[5pt]
&\frac{5}{6}=\frac {1}{2}+\frac {1}{3}-P(X\cap Y)\\[5pt]
P(X\cap Y)&=\frac {1}{2}+\frac {1}{3}-\frac {5}{6}\\[5pt]
&=\frac {3+2-5}{6}\\[5pt]
&=\frac {0}{6}\\[5pt]
&=\underline{\underline{0}}\\[5pt]
(ii)P(X\cap Y)&=0\text { }\text{ , X and Y are mutually exclusive} \\[5pt]
\end{align}$
| | |
 |
$\begin{align}&P(X)=\frac{1}{6} , P(Y)=\frac{1}{9} , P(Z')=\frac{2}{3} , P(X\cap Y)=\frac{1}{18} , P(X\cap Z)=\frac{1}{12} \\[5pt]
(i)P(X')&=1-P(X) \\[5pt]
&=1-\frac {1}{6}\\[5pt]
&=\underline{\underline{\frac{5}{6}}}
\end{align}$
| $\begin{align}(ii)P(Y')&=1-P(Y) \\[5pt]
&=1-\frac {1}{9}\\[5pt]
&=\underline{\underline{\frac{8}{9}}}
\end{align}$
| $\begin{align}(iii)P(Z)&=1-P(Z') \\[5pt]
&=1-\frac {2}{3}\\[5pt]
&=\underline{\underline{\frac{1}{3}}}
\end{align}$
|
$\begin{align}(iv)P(X\cup Y)&=P(X)+P(Y)-P(X\cap Y) \\[5pt]
&=\frac {1}{6}+\frac {1}{9}-\frac {1}{18}\\[5pt]
&=\frac {3+2-1}{18}\\[5pt]
&=\frac {4}{18}\\[5pt]
&=\underline{\underline{\frac{2}{9}}}
\end{align}$
| $\begin{align}(v)P(X\cup Z)&=P(X)+P(Z)-P(X\cap Y) \\[5pt]
&=\frac {1}{6}+\frac {1}{3}-\frac {1}{12}\\[5pt]
&=\frac {2+4-1}{12}\\[5pt]
&=\underline{\underline{\frac{5}{12}}}\\[5pt]
P[(X\cup Z)']&=1-P(X\cup Z)\\[5pt]
&=1-\frac {5}{12}\\[5pt]
&=\underline{\underline{\frac{7}{12}}}
\end{align}$
| |