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 $3 : 5 : 2$ , respectively. The germination rates of the three types of seeds are $40 %, 60 %$ and $40 %$ respectively. Calculate the probability of a randomly chosen seed to germinate.

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Solution

The mixture of $A_{1}, A_{2}, A_{3}$ type seeds is $3 : 5 : 2$ .
We have, $P (A_{1}) = \frac{3}{10} = 0.3 P (A_{2}) = \frac{5}{10} = 0.5 P (A_{3}) = \frac{2}{10} = 0.2$

Let $E$ be the event that a seed germinates, then
$P (E | A_{1}) = 0.4 P (E | A_{2}) = 0.6 P (E | A_{3}) = 0.4$

By using 'Theorem of total probability', we have
$P (E) = P (A_{1}) \times P (E | A_{1}) + P (A_{2}) \times P (E | A_{2}) + P (A_{3}) \times P (E | A_{3})$
$= 0.3 \times 0.4 + 0.5 \times 0.6 + 0.2 \times 0.4$
$= 0.5$

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A shopkeeper sells three types of flower seeds A1,A2 and A3. They are sold as a mixture, where the proportions are 3:5:2, respectively. The germination rates of the three types of seeds are 40%,60% and 40% respectively. Calculate the probability of a randomly chosen seed to germinate.