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.

Answer 1

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