A shopkeeper sells three types of flower seeds $A_{1}, A_{2}$ and $A_{3}$ . They are sold as a mixture, where the proportions are 4 : 4 : 2, respectively. The germination rates of the three types of seeds are 45%, 60% and 35% . Calculate the probability

(i) of a randomly chosen seed to germinate.

(ii) that it will not germinate given that the seed is of type $A_{3}$ .

(iii) that it is of the type $A_{2}$ given that a randomly chosen seed does not germinate.

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Solution

We have $A_{1} : A_{2} : A_{3}$ = 4 : 4 : 2

$P (A_{1}) = \frac{4}{10}, P (A_{2}) = \frac{4}{10} a n d P (A_{3}) = \frac{2}{10}$

where $A_{1}, A_{2}$ and $A_{3}$ denote the three types of flower seeds.

Let E be the event that a seed germinales and $¯ E$ be the event that a seed does not germinate.

$∴ P (E / A_{1}) = \frac{45}{100}, P (E / A_{2}) = \frac{60}{100} a n d P (E / A_{3}) = \frac{35}{100}$

and $P (¯ E / A_{1}) = \frac{55}{100}, P (¯ E / A_{2}) = \frac{40}{100} a n d P (¯ E / A_{3}) = \frac{65}{100}$

(i) $∴$ P(E) = $P (A_{1}) P (E / A_{1})$ + $P (A_{2}) P (E / A_{2}) + P (A_{3}) P (E / A_{3})$

$= \frac{4}{10} \frac{45}{100} + \frac{4}{10} \frac{60}{100} + \frac{2}{10} \frac{35}{100}$

$= \frac{180}{1000} + \frac{240}{1000} + \frac{70}{1000} = 0.49$

(ii) $P (¯ E / A_{3})$ = 1 - $P (E / A_{3})$ = 1 - $\frac{35}{100} = \frac{65}{100}$

(iii) $P (A_{2} / ¯ E) = \frac{P (A_{2}) . P (¯ E / A_{2})}{P (A_{1}) . P (¯ E / A_{1}) + P (A_{2}) . P (¯ E / A_{2}) + P (A_{3}) . P (¯ E / A_{3})}$
$= \frac{\frac{4}{10} . \frac{40}{100}}{\frac{4}{10} . \frac{55}{100} + \frac{4}{10} . \frac{40}{100} + \frac{2}{10} . \frac{65}{100}} = \frac{\frac{160}{1000}}{\frac{220}{1000} + \frac{160}{1000} + \frac{130}{1000}}$
$= \frac{160 / 1000}{510 / 1000} = \frac{16}{51} = 0.313725 = 0.314$

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