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 of type \(A_1\) given that seed doesn’t germinate.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{22}{51}\)
Let G be the event that the seed germinates. With \(P(A_1) = P(A_2) = 0.4\), \(P(A_3) = 0.2\) and germination rates 0.45, 0.60, 0.35:
\(P(G) = 0.18 + 0.24 + 0.07 = 0.49\)
\(P(G') = 1 - 0.49 = 0.51\)
A seed of type \(A_1\) fails to germinate with probability 1 − 0.45 = 0.55. By Bayes’ theorem:
\( \displaystyle P(A_1 \mid G') = \frac{0.4 \times 0.55}{0.51} \)
\( \displaystyle = \frac{0.22}{0.51} = \frac{22}{51} \)