Question

A gardener wanted to plant vegetables in his garden. Hence he bought 10 seeds of brinjal plant, 12 seeds of cabbage plant, and 8 seeds of radish plant. The shopkeeper assured him of germination probabilities of brinjal, cabbage, and radish to be 25%, 35%, and 40% respectively. But before he could plant the seeds, they got mixed up in the bag and he had to sow them randomly.

What is the probability that it is a cabbage seed, given that the chosen seed germinates?

Hint:

To find conditional probability, divide the probability of the event happening with both conditions (germination and cabbage) by the total probability of germination.

Answer 1

The objective is to determine the conditional probability of a seed being a cabbage seed, given that it germinates. This is calculated using the conditional probability formula: \[ P(\text{Cabbage seed} \mid \text{Germinate}) = \frac{P(\text{Cabbage seed and Germinate})}{P(\text{Germinate})} \] We are given \( P(\text{Germinate}) \approx 0.267 \). The probability of a seed being both a cabbage seed and germinating is: \[ P(\text{Cabbage seed and Germinate}) = P(\text{Cabbage}) \cdot P(\text{Cabbage seed}) = 0.35 \cdot \frac{2}{5} = 0.14 \] Therefore, the conditional probability is: \[ P(\text{Cabbage seed} \mid \text{Germinate}) = \frac{0.14}{0.267} \approx 0.523 \] The probability that a germinating seed is a cabbage seed is approximately \( \boxed{0.523} \).