$\begin{align}(i)&P(A)=\frac {n(A)}{n(S)}\\[5pt] &\underline{\underline{=\frac{23} {50}}} \end{align}$	$\begin{align}(ii)&P(A')=1-P(A)\\[5pt] =&1-\frac {23}{50}\\[5pt] =&\frac {50-23}{50}\\[5pt] &\underline{\underline{=\frac{27} {50}}} \end{align}$
$\begin{align}&S=\{1,2,3,4,5,6,7,8\}\\[5pt] &(i)\text{A has 8 events.}A _{1}, A _{2}, A _{3}, A _{4}, A _{5}, A _{6}, A _{7}, A _{8}\\[5pt] &A _{1}=\{1\} , A _{2}=\{2\} , A _{3}=\{3\} , A _{4}=\{4\}\\[5pt] &A _{5}=\{5\} , A _{6}=\{6\} , A _{7}=\{7\} , A _{8}=\{8\}\\[5pt] &(ii)\\[5pt] &n ({A_1})=1 , n ({A_2})=1 , n ({A_3})=1 , n ({A_4})=1\\[5pt] &n ({A_5})=1 , n ({A_6})=1 , n ({A_7})=1 , n ({A_8})=1 , n ({S})=8\\[5pt] &P ({A_1})=\frac{n (A_1)}{n (S)}=\frac{1}{8} , P ({A_2})=\frac{n (A_2)}{n (S)}=\frac{1}{8} ,P ({A_3})=\frac{n (A_3)}{n (S)}=\frac{1}{8} ,P ({A_4})=\frac{n (A_4)}{n (S)}=\frac{1}{8} \\[5pt] &P ({A_5})=\frac{n (A_5)}{n (S)}=\frac{1}{8} , P ({A_6})=\frac{n (A_6)}{n (S)}=\frac{1}{8} ,P ({A_7})=\frac{n (A_7)}{n (S)}=\frac{1}{8} ,P ({A_8})=\frac{n (A_8)}{n (S)}=\frac{1}{8} ,\\[5pt] &=\underline{\underline{P (A)=\frac{n (A)}{n (S)}=\frac{1}{8}}}\\[5pt] &(iii)B=\{1,3,5,7\}\\[5pt] &(iv)P ({B})=\frac{n (B)}{n (S)}=\frac{4}{8}=\frac{1}{2}\\[5pt] &P ({B'})=1-P(B)=1-\frac{1}{2}=\frac{1}{2}\\[5pt] &(v)P ({X})=0.5=\frac{1}{2}=\frac{4}{8}\\[5pt] &X_1 =\{2,4,6,8\}\\[5pt] &X_2 =\{2,3,5,7\}\\[5pt] \end{align}$
$\begin{align}&(i)n(S)=12+3+7+5=27\\[5pt] &(ii)P (A)=\frac{n (A)}{n (S)}=\frac{15}{27}=\frac{5}{9}\\[5pt] &(iii)P (B)=\frac{n (B)}{n (S)}=\frac{10}{27}\\[5pt] &(iv)P (A\cap B)=\frac{n (A\cap B)}{n (S)}=\frac{3}{27}=\frac{1}{9}\\[5pt] &(v)P (A\cup B)=\frac{n (A\cup B)}{n (S)}=\frac{22}{27}\\[5pt] &(vi)P (A\cap B')=\frac{n (A\cap B')}{n (S)}=\frac{12}{27}=\frac{4}{9}\\[5pt] &(vii)P (A'\cap B)=\frac{n (A'\cap B)}{n (S)}=\frac{7}{27}\\[5pt] &(viii)P (A\cup B)'=\frac{n (A\cup B)'}{n (S)}=\frac{5}{27}\\[5pt] \end{align}$
$\begin{align}&(i)S=\{(1,1) , (1,2) , (1,3) , (2,1) , (2,2) , (2,3) , (3,1) , (3,2) , (3,3)\}\\[5pt] &n(S)=9\\[5pt]\\[5pt] &(ii)A=\{(2,2)\}\\[5pt] &n(A)=1\\[5pt]\\[5pt] &(iii)P(A)=\frac {n(A)}{n(S)}\\[5pt] &p(A)=\frac {1}{9}\\[5pt]\\[5pt] \end{align}$ $\begin{align}&(iv)\\[5pt] \end{align}$ $\begin{align}& \\[5pt] &(v)P(B)=\frac {n(B)}{n(S)}\\[5pt] &=\frac {4}{9}\\[5pt]\\[5pt] &(vi)\\[5pt] \end{align}$ $\begin{align}& \\[5pt] &\text {Probability of drawing at least one even number =P(o,e)+P(e,o)+P(e,e)}\\[5pt] &=\frac {2}{9} + \frac {2}{9} + \frac {1}{9}\\[5pt] &=\frac {5}{9}\\[5pt] \end{align}$

By studying this lesson you will be able to

Solve problems involving the events of a random experiment occurring in two stages using

1. A grid (Cartesian plane)
2. a tree diagram