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# 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$\begin{array}{1 1}(i)\;of\;a\;randomly\;chosen\;seed\;to\;germinate\$ii)\;that\;it\;will\;not\;germinate\;given\;that\;seed\;is\;of\;type\;A_3\\(iii)\;that\;it\;is\;of\;the\;type\;A_2\;given\;that\;a\;randomly\;chosen\;seed\;does\;not\;germinate\end{array}$ Can you answer this question? ## 1 Answer 0 votes 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)
=$\Large\frac{\frac{2}{5}\times\frac{40}{100}}{\frac{2}{5}\times\frac{55}{100}+\frac{2}{5}\times\frac{40}{100}+\frac{1}{5}\times\frac{65}{100}}$
=$\Large\frac{\frac{4}{25}}{\frac{110+80+65}{500}}$
=$0.314$

edited Jun 4, 2013