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Question
A shopkeeper sells three types of flower seeds A1, A2 and A3. 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 that it is of the type A2 given that a randomly chosen seed does not germinate.
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Solution
Given that A1: A2: A3 = 4: 4: 2
∴ P(A1) = `4/10`
P(A2) = `4/10`
And P(A3) = `2/10`
Where A1, A2 and A3 are the three types of seeds.
Let E be the event that a seed germinates and `bar"E"` be the event that a seed does not germinate
∴ `"P"("E"/"A"_1) = 45/100 "P"("E"/"A"_2) = 60/100` and `"P"("E"/"A"_3) = 35/100`
And `"P"(bar"E"/"A"_1) = 55/100, "P"(bar"E"/"A"_2) = 40/100` and `"P"(bar"E"/"A"_3) = 65/100`
Using Bayes’ Theorem, we get
`"P"("A"_2/bar"E") = ("P"("A"_2)*"P"(bar"E"/"A"_2))/("P"("A"_1)*"P"(bar"E"/"A"_1) + "P"("A"_2)*"P"(bar"E"/"A"_2) + "P"("A"_3)*"P"(bar"E"/"A"_3))`
= `(4/10*40/100)/(4/10*55/100 + 4/10*40/100 + 2/10*65/100)`
= `(160/1000)/(220/1000 + 160/1000 + 130/1000)`
= `160/(220 + 160 + 130)`
= `160/510`
= `16/51`
= 0.314
Hence, the required probability is `16/51` or 0.314
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