# A shop keeper sells three types of flower seeds $A_1,A_2 and\;A_3$.They are sold as a mixture where the proportions arev4:4:2 respectively.The germination rates of the three types of seeds are 45%,60% and 35%.Calculate the probability$\begin{array}{1 1}(i)\;of\;a\;randomly\;chosen\;seed\;to\;germinate\$$ii)\;that\;it\;will\;not\;germinate\;given\;that\;seed\;is\;of\;type\;A_3\\(iii)\;that\;it\;is\;of\;the\;type\;A_2\;given\;that\;a\;randomly\;chosen\;seed\;does\;not\;germinate\end{array}$ ## 1 Answer Toolbox: • The mixture of \(A_1\;A_2\;A_3$$ type seeds is $$4:\;4:\;2$$
• Let $$E_1 \;seeds \;selected \;is\;A_1$$
• Let $$E_2 \;seeds \;selected \;is\;A_2$$
• Let $$E_3\;seeds \;selected \;is\;A_3$$
• $$A\;the\; seeds\;germinates$$
• P(A)=P($$E_1$$)P($$A/E_1$$)+p($$E_2$$)P($$A/E_2$$)+P($$E_3$$)P($$A/E_3$$)
• P($$\bar{A}/E_1$$)=1-P(A/$$E_1$$)
• P($$\bar{A}/E_2$$)=1-P(A/$$E_2$$)
• P($$\bar{A}/E_3$$)=1-P(A/$$E_3$$)
• P($$E_2\bar{A}$$)=$$\Large\frac{{P(E_2)}{P(\bar{A}/E_2)}}{{p(E_1)}{p(\bar{A}/E_1)}+{P(E_2)}{P(\bar{A}/E_2)}+{(P(E_3)}{P(\bar{A}/E_3)}}$$
P($$E_1$$)=$$\large\frac{4}{10}$$=$$\frac{2}{5}$$ $$seeds\; 'A'\;is\; settled$$
P($$E_2$$)=$$\large\frac{4}{10}$$=$$\frac{2}{5}$$ $$seeds\; 'A_2'\;is\; settled$$
P($$E_3$$)=$$\large\frac{2}{10}$$=$$\frac{1}{5}$$ $$seeds\; 'A_3'\;is\; settled$$
P($$A/E_1$$)=P(germinates/$$A_1$$ selected)
=$$\large\frac{45}{100}$$
P($$\bar{A}/E_1$$)=$$\frac{55}{100}$$
P($$A/E_2$$)=P(germinates/$$A_2$$ selected)
=$$\large\frac{60}{100}$$
P($$\bar{A}/E_2$$)=$$\large\frac{40}{100}$$
P($$A/E_3$$)=P(germinates/$$A_3$$ selected)
=$$\large\frac{35}{100}$$
P($$\bar{A}/E_3$$)=$$\large\frac{65}{100}$$
P(A)=P(germinates)
=$$\Large\frac{2}{5}$$$$\times$$$$\frac{45}{100}$$+$$\frac{2}{5}$$$$\times$$$$\frac{60}{100}$$+$$\frac{1}{5}$$$$\times$$$$\frac{35}{100}$$
=$$\large\frac{245}{500}$$=0.49
P($$\bar{A}/E_3$$)=P(does not germinate/seed $$A_3$$ is selected)
=$$1-\large\frac{35}{100}$$=$$\large\frac{65}{100}$$
=$$0.65$$
P($$E_2/\bar{A}$$)=P(seeds$$A_2$$ is chosen/the seeds does not germinate)
=$$\Large\frac{\frac{2}{5}\times\frac{40}{100}}{\frac{2}{5}\times\frac{55}{100}+\frac{2}{5}\times\frac{40}{100}+\frac{1}{5}\times\frac{65}{100}}$$
=$$\Large\frac{\frac{4}{25}}{\frac{110+80+65}{500}}$$
=$$0.314$$

answered Mar 2, 2013
edited Jun 4, 2013

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