Question Number 134623 by bobhans last updated on 05/Mar/21
$$\mathscr{PR}\mathrm{obability} \\ $$ An urn contains 3 red balls, 2 green balls and 1 yellow ball. Three balls are selected at random and without replacement from the urn. What is the probability that at least 1 color is not drawn? by : Bobhans\\n
Answered by EDWIN88 last updated on 05/Mar/21
$$\mathrm{Total}\:\mathrm{of}\:\mathrm{balls}\:=\:\mathrm{3}+\mathrm{2}+\mathrm{1}\:=\:\mathrm{6}\: \\ $$ $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{selecting}\:\mathrm{3}\:\mathrm{balls}\:\mathrm{from} \\ $$ $$\mathrm{these}\:\mathrm{6}\:\mathrm{balls}\:=\:\begin{pmatrix}{\mathrm{6}}\\{\mathrm{3}}\end{pmatrix}\:=\:\mathrm{\color{mathred}{2}\color{mathred}{0}} \\ $$ $$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{selecting}\:\mathrm{1}\:\mathrm{ball}\:\mathrm{of} \\ $$ $$\mathrm{each}\:\mathrm{color}\:=\:\begin{pmatrix}{\mathrm{3}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}×\begin{pmatrix}{\mathrm{1}}\\{\mathrm{1}}\end{pmatrix}\:=\:\mathrm{\color{mathred}{6}} \\ $$ $$\mathrm{\color{mathbrown}{T}\color{mathbrown}{h}\color{mathbrown}{e}\color{mathbrown}{r}\color{mathbrown}{e}\color{mathbrown}{f}\color{mathbrown}{o}\color{mathbrown}{r}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{p}\color{mathbrown}{r}\color{mathbrown}{o}\color{mathbrown}{b}\color{mathbrown}{a}\color{mathbrown}{b}\color{mathbrown}{i}\color{mathbrown}{l}\color{mathbrown}{i}\color{mathbrown}{t}\color{mathbrown}{y}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{o}\color{mathbrown}{f}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{s}\color{mathbrown}{e}\color{mathbrown}{l}\color{mathbrown}{e}\color{mathbrown}{c}\color{mathbrown}{t}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{g}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{3}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{b}\color{mathbrown}{a}\color{mathbrown}{l}\color{mathbrown}{l}\color{mathbrown}{s}} \\ $$ $$\mathrm{\color{mathbrown}{o}\color{mathbrown}{f}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{d}\color{mathbrown}{i}\color{mathbrown}{f}\color{mathbrown}{f}\color{mathbrown}{e}\color{mathbrown}{r}\color{mathbrown}{e}\color{mathbrown}{n}\color{mathbrown}{t}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{c}\color{mathbrown}{o}\color{mathbrown}{l}\color{mathbrown}{o}\color{mathbrown}{r}}\color{mathbrown}{\:}\color{mathbrown}{=}\mathrm{\color{mathbrown}{p}}\color{mathbrown}{\left(}\mathrm{\color{mathbrown}{A}}\color{mathbrown}{\right)}\color{mathbrown}{=}\frac{\mathrm{\color{mathbrown}{6}}}{\mathrm{\color{mathbrown}{2}\color{mathbrown}{0}}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\frac{\mathrm{\color{mathbrown}{3}}}{\mathrm{\color{mathbrown}{1}\color{mathbrown}{0}}} \\ $$ $$\mathrm{\color{mathred}{s}\color{mathred}{o}}\color{mathred}{\:}\mathrm{\color{mathred}{t}\color{mathred}{h}\color{mathred}{e}}\color{mathred}{\:}\mathrm{\color{mathred}{p}\color{mathred}{r}\color{mathred}{o}\color{mathred}{b}\color{mathred}{a}\color{mathred}{b}\color{mathred}{i}\color{mathred}{l}\color{mathred}{i}\color{mathred}{t}\color{mathred}{y}}\color{mathred}{\:}\mathrm{\color{mathred}{t}\color{mathred}{h}\color{mathred}{a}\color{mathred}{t}}\color{mathred}{\:}\mathrm{\color{mathred}{l}\color{mathred}{e}\color{mathred}{a}\color{mathred}{s}\color{mathred}{t}}\color{mathred}{\:}\mathrm{\color{mathred}{1}}\color{mathred}{\:}\mathrm{\color{mathred}{c}\color{mathred}{o}\color{mathred}{l}\color{mathred}{o}\color{mathred}{r}}\color{mathred}{\:} \\ $$ $$\mathrm{\color{mathred}{i}\color{mathred}{s}}\color{mathred}{\:}\mathrm{\color{mathred}{n}\color{mathred}{o}\color{mathred}{t}}\color{mathred}{\:}\mathrm{\color{mathred}{d}\color{mathred}{r}\color{mathred}{a}\color{mathred}{w}\color{mathred}{n}}\color{mathred}{\:}\mathrm{\color{mathred}{i}\color{mathred}{s}}\color{mathred}{\:}\mathrm{\color{mathred}{p}}\color{mathred}{\left(}\mathrm{\color{mathred}{A}}^{\mathrm{\color{mathred}{c}}} \color{mathred}{\right)}\color{mathred}{\:}\color{mathred}{=}\mathrm{\color{mathred}{1}}\color{mathred}{−}\mathrm{\color{mathred}{p}}\color{mathred}{\left(}\mathrm{\color{mathred}{A}}\color{mathred}{\right)}\color{mathred}{=}\frac{\mathrm{\color{mathred}{7}}}{\mathrm{\color{mathred}{1}\color{mathred}{0}}} \\ $$