Question 14 Marks
Read the text carefully and answer the questions:
A shopkeeper sells three types of flower seeds $\mathrm{A}_{1}, \mathrm{~A}_{2}$, and $\mathrm{A}_{3}$. 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 $35 \%$ respectively.
(a) Calculate the probability of randomly chosen seed to germinate.
(b) Calculate the probability that it is of the type $\mathrm{A}_{2}$ given that randomly chosen seed does not germinate.
(c) Calculate the probability that it will not germinate given that the seed is of type $\mathrm{A}_{1}$.
A shopkeeper sells three types of flower seeds $\mathrm{A}_{1}, \mathrm{~A}_{2}$, and $\mathrm{A}_{3}$. 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 $35 \%$ respectively.
(a) Calculate the probability of randomly chosen seed to germinate.
(b) Calculate the probability that it is of the type $\mathrm{A}_{2}$ given that randomly chosen seed does not germinate.
(c) Calculate the probability that it will not germinate given that the seed is of type $\mathrm{A}_{1}$.
Answer
View full question & answer→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 $35 \%$ respectively.
(i) Here, $\mathrm{P}\left(\mathrm{A}_{1}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{2}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{3}\right)=\frac{2}{10}$,
and $\mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)=\frac{45}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)=\frac{60}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)=\frac{35}{100}$
where $G$ is the event that seeds germinate.
$\mathrm{P}(\mathrm{G})=\mathrm{P}\left(\mathrm{A}_{1}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)+\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)+\mathrm{P}\left(\mathrm{A}_{3}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)$
$=\frac{4}{10} \times \frac{45}{100}+\frac{4}{10} \times \frac{60}{100}+\frac{2}{10} \times \frac{35}{100}=\frac{490}{1000}=0.49$
(ii) Here, $\mathrm{P}\left(\mathrm{A}_{1}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{2}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{3}\right)=\frac{2}{10}$,
and $\mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)=\frac{45}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)=\frac{60}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)=\frac{35}{100}$
where $G$ is the event that seeds germinate.
Required probability $=P\left(\mathrm{~A}_{2} \mid \mathrm{G}^{\prime}\right)=\frac{\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{2}\right)}{\mathrm{P}\left(\mathrm{A}_{1}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{1}\right)+\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{2}\right)+\mathrm{P}\left(\mathrm{A}_{3}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{3}\right)}$
$
=\frac{\frac{4}{10} \times \frac{40}{100}}{\frac{4}{10} \times \frac{55}{100}+\frac{4}{10} \times \frac{40}{100}+\frac{2}{10} \times \frac{65}{100}}=\frac{160}{510}=\frac{16}{51}=0.314
$
(iii) P (seeds of type $\mathrm{A}_{1}$ will not germinate) $=1-\frac{45}{100}=\frac{55}{100}$.
(i) Here, $\mathrm{P}\left(\mathrm{A}_{1}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{2}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{3}\right)=\frac{2}{10}$,
and $\mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)=\frac{45}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)=\frac{60}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)=\frac{35}{100}$
where $G$ is the event that seeds germinate.
$\mathrm{P}(\mathrm{G})=\mathrm{P}\left(\mathrm{A}_{1}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)+\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)+\mathrm{P}\left(\mathrm{A}_{3}\right) \cdot \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)$
$=\frac{4}{10} \times \frac{45}{100}+\frac{4}{10} \times \frac{60}{100}+\frac{2}{10} \times \frac{35}{100}=\frac{490}{1000}=0.49$
(ii) Here, $\mathrm{P}\left(\mathrm{A}_{1}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{2}\right)=\frac{4}{10}, \mathrm{P}\left(\mathrm{A}_{3}\right)=\frac{2}{10}$,
and $\mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{1}\right)=\frac{45}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{2}\right)=\frac{60}{100}, \mathrm{P}\left(\mathrm{G} \mid \mathrm{A}_{3}\right)=\frac{35}{100}$
where $G$ is the event that seeds germinate.
Required probability $=P\left(\mathrm{~A}_{2} \mid \mathrm{G}^{\prime}\right)=\frac{\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{2}\right)}{\mathrm{P}\left(\mathrm{A}_{1}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{1}\right)+\mathrm{P}\left(\mathrm{A}_{2}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{2}\right)+\mathrm{P}\left(\mathrm{A}_{3}\right) \cdot \mathrm{P}\left(\mathrm{G}^{\prime} \mid \mathrm{A}_{3}\right)}$
$
=\frac{\frac{4}{10} \times \frac{40}{100}}{\frac{4}{10} \times \frac{55}{100}+\frac{4}{10} \times \frac{40}{100}+\frac{2}{10} \times \frac{65}{100}}=\frac{160}{510}=\frac{16}{51}=0.314
$
(iii) P (seeds of type $\mathrm{A}_{1}$ will not germinate) $=1-\frac{45}{100}=\frac{55}{100}$.
