Resources for Complementary Events
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Complementary Events Theory
![Complementary events are events that together make up all the possible outcomes and the sum of their probabilities must equal 1. \\ The complement of an event \(E\) is usually written as \(\bar{E}\) (not \(E\) ).\\ \(\mathrm{P}(\) event \()+\mathrm{P}(\) complementary event \()=1\)\\ $\begin{aligned} P(E)+P(\bar{E}) &=1 \\ P(\bar{E}) &=1-P(E) \text{where } \bar{E} \text{ is the complement of }E \end{aligned}$ \\ \begin{multicols}{2} \textbf{Example}\\ The spinner is spun.\\ \begin{center} \begin{tikzpicture} \draw[line width=1.5pt,fill=white!20!white] (35:2cm)--(107:2cm)--(179:2cm)--(251:2cm)--(323:2cm)--(35:2cm)--cycle; \draw[line width=1.5pt,black] (35:2cm)--(107:2cm)--(179:2cm)--(251:2cm)--(323:2cm)--(35:2cm)--cycle; {\foreach \x in {35,107,179,251,323} \draw[line width=1.5pt,black] (0,0)--(\x:2cm); } \foreach \y/\label in {71/5,142/1,213/2,284/3,355/4} \node[] at (\y:1cm) {\Large \label}; \fill[black] (0,0) circle (5pt); \draw[line width=0.1pt,fill=black] (-90:4.5pt)--(2:1.5cm)--(90:4.5pt); \fill[white] (0,0) circle (2pt); \end{tikzpicture} %\includegraphics[width=0.3\textwidth]{2ecc6cd7-73f2-4cdb-8122-1e856bdf3d85} \end{center} a) List the sample space.\\ b) Find \(P(4)\).\\ c) List the outcomes in the complementary event of spinning a 4.\\ d) If \(P(\operatorname{not} 4)=P(\overline{4})\), find \(P(\overline{4})\).\\ \columnbreak \textbf{Solution}\\ a) Sample space \(=\{1,2,3,4,5\}\)\\ b) \(P(4)=\dfrac{1}{5}\) ( 1 chance in 5)\\ c) The outcomes in the complementary event are \(\{1,2,3,5\}\).\\ d) \(P(\overline{4})=1-P(4)=1-\dfrac{1}{5}=\dfrac{5-1}{5}=\dfrac{4}{5}\)\\ \end{multicols}](/media/1e2imkpc/9243.png)
NSW Y8 Maths Probability Complementary Events
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