A shopkeeper sells three types of flower seeds \(A_1\), \(A_2\), \(A_3\). 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. Calculate the probability in the following cases.
The probability that seed is not of type \(A_1\), given that seed germinates.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{31}{49}\)
Let G be the event that the seed germinates. The three parts of the total probability of germinating are
\(P(A_1 \cap G) = 0.4 \times 0.45 = 0.18\)
\(P(A_2 \cap G) = 0.4 \times 0.60 = 0.24\)
\(P(A_3 \cap G) = 0.2 \times 0.35 = 0.07\)
so P(G) = 0.49. By Bayes’ theorem:
\( \displaystyle P(A_1 \mid G) = \frac{0.18}{0.49} = \frac{18}{49} \)
Not of type \(A_1\) is the complementary event:
\( \displaystyle P(A_1' \mid G) = 1 - \frac{18}{49} = \frac{31}{49} \)