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Home  >>  CBSE XII  >>  Math  >>  Probability
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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\[\begin{array}{1 1}(i)\;of\;a\;randomly\;chosen\;seed\;to\;germinate\\(ii)\;that\;it\;will\;not\;germinate\;given\;that\;seed\;is\;of\;type\;A_3\\(iii)\;that\;it\;is\;of\;the\;type\;A_2\;given\;that\;a\;randomly\;chosen\;seed\;does\;not\;germinate\end{array}\]

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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 by poojasapani_1
edited Jun 4, 2013 by poojasapani_1
 

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