Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 185203 by KONE last updated on 18/Jan/23

Answered by aba last updated on 18/Jan/23

$$$$$$1.\mathrm{P}\left({\mathrm{G}}_1\right)=\mathrm{P}\left({\mathrm{G}}_1/{\mathrm{A}}_1\right)\times \mathrm{P}\left({\mathrm{A}}_1\right)+\mathrm{P}\left({\mathrm{G}}_1/{\stackrel{-}{\mathrm{A}}}_1\right)\times \mathrm{P}\left({\stackrel{-}{\mathrm{A}}}_1\right)$$$$=\frac{1}{5}\times \frac{1}{2}+\frac{1}{10}\times (1-\frac{1}{2})$$$$=\frac{3}{20}$$$$2.\mathrm{Make}\mathrm{a}\mathrm{tree}\mathrm{P}\left({\mathrm{G}}_2\right)=\frac{1}{2}(\frac{1}{5}\times \frac{1}{5}+\frac{4}{5}\times \frac{1}{10})+\frac{1}{2}(\frac{1}{10}\times \frac{1}{10}+\frac{9}{10}\times \frac{1}{5})=\frac{31}{200}$$$$3.{\mathrm{let}}^{\prime }\mathrm{s}\mathrm{use}\mathrm{bayes}\mathrm{formula}$$$$\mathrm{P}\left({\mathrm{A}}_1/{\mathrm{G}}_2\right)=\frac{\mathrm{P}({\mathrm{A}}_1\cap {\mathrm{G}}_2)}{\mathrm{P}\left({\mathrm{G}}_2\right)}=\frac{\mathrm{P}\left({\mathrm{G}}_2/{\mathrm{A}}_1\right)\mathrm{P}\left({\mathrm{A}}_1\right)}{\mathrm{P}\left({\mathrm{G}}_2\right)}=\frac{\frac{3}{25}\times \frac{1}{5}}{\frac{31}{200}}=\frac{12}{31}$$

Answered by aba last updated on 18/Jan/23

$$4.1\mathrm{P}\left({\mathrm{G}}_{\mathrm{k}}\right)=\mathrm{P}\left({\mathrm{G}}_{\mathrm{k}}/{\mathrm{A}}_{\mathrm{k}}\right)\times \mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)+\mathrm{P}\left({\mathrm{G}}_{\mathrm{k}}/{\stackrel{-}{\mathrm{A}}}_{\mathrm{k}}\right)\times \mathrm{P}\left({\stackrel{-}{\mathrm{A}}}_{\mathrm{k}}\right)$$$$=\frac{1}{5}\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)+\frac{1}{10}(1-\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right))$$$$=\frac{1}{10}(1+\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right))$$$$4.2\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}+1}\right)=\mathrm{P}({\mathrm{G}}_{\mathrm{k}}\cap {\mathrm{A}}_{\mathrm{k}})+\mathrm{P}({\stackrel{-}{\mathrm{G}}}_{\mathrm{k}}\cap {\stackrel{-}{\mathrm{A}}}_{\mathrm{k}})$$$$=\mathrm{P}\left({\mathrm{G}}_{\mathrm{k}}/{\mathrm{A}}_{\mathrm{k}}\right)\times \mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)+\mathrm{P}\left({\stackrel{-}{\mathrm{G}}}_{\mathrm{k}}/{\stackrel{-}{\mathrm{A}}}_{\mathrm{k}}\right)\times \mathrm{P}\left({\stackrel{-}{\mathrm{A}}}_{\mathrm{k}}\right)$$$$=\frac{1}{5}\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)+\frac{9}{10}(1-\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right))$$$$=\frac{9}{10}-\frac{7}{10}\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)$$$$4.3\mathrm{geometric}\mathrm{arithmetic}\mathrm{sequence}({\mathrm{U}}_{\mathrm{n}+1}={\mathrm{aU}}_{\mathrm{n}}+\mathrm{b})$$$$\mathrm{let}\mathrm{l}=\frac{9}{10}-\frac{7}{10}\mathrm{l}\Rightarrow \mathrm{l}=\frac{9}{17}$$$${\mathrm{v}}_{\mathrm{k}}=\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)-\frac{9}{17}\Rightarrow {\mathrm{v}}_{\mathrm{k}+1}=\frac{-7}{10}{\mathrm{v}}_{\mathrm{k}}$$$$\Rightarrow {\mathrm{v}}_{\mathrm{k}}={\left(\frac{-7}{10}\right)}^{\mathrm{k}-1}{\mathrm{v}}_1$$$$\mathrm{P}\left({\mathrm{A}}_{\mathrm{k}}\right)={(-\frac{7}{10})}^{\mathrm{k}-1}(\mathrm{P}\left({\mathrm{A}}_1\right)-\frac{9}{17})+\frac{9}{17}$$$$=-\frac{1}{34}{(-\frac{7}{10})}^{\mathrm{k}-1}+\frac{9}{17}$$$$\mathrm{we}\mathrm{then}\mathrm{deduce}:$$$$\mathrm{P}\left({\mathrm{G}}_{\mathrm{k}}\right)=\frac{13}{85}-\frac{1}{340}{(-\frac{7}{10})}^{\mathrm{k}-1}$$

Commented by KONE last updated on 20/Jan/23

$$thankyousir$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com