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.

Calculate the probability of a randomly chosen seed to germinate.

Hint:

For calculating the total probability, use the law of total probability. Multiply the probability of each event by the probability of selecting each type of seed.

Answer 1

We are given the following information: - There are 10 brinjal seeds, 12 cabbage seeds, and 8 radish seeds, making a total of \( 10 + 12 + 8 = 30 \) seeds. - The probability of germination for each type of seed is: - Brinjal: \( P(\text{Brinjal}) = 0.25 \) - Cabbage: \( P(\text{Cabbage}) = 0.35 \) - Radish: \( P(\text{Radish}) = 0.40 \) We need to calculate the total probability of a randomly chosen seed germinating. This is given by the law of total probability: \[ P(\text{Germinate}) = P(\text{Brinjal}) \cdot P(\text{Brinjal seed}) + P(\text{Cabbage}) \cdot P(\text{Cabbage seed}) + P(\text{Radish}) \cdot P(\text{Radish seed}) \] The probabilities of choosing each seed are: - \( P(\text{Brinjal seed}) = \frac{10}{30} = \frac{1}{3} \) - \( P(\text{Cabbage seed}) = \frac{12}{30} = \frac{2}{5} \) - \( P(\text{Radish seed}) = \frac{8}{30} = \frac{4}{15} \) Substitute these into the formula: \[ P(\text{Germinate}) = 0.25 \cdot \frac{1}{3} + 0.35 \cdot \frac{2}{5} + 0.40 \cdot \frac{4}{15} \] Now, calculate each term: \[ P(\text{Germinate}) = \frac{0.25}{3} + \frac{0.70}{5} + \frac{1.60}{15} \] Finally, adding these values: \[ P(\text{Germinate}) = \frac{0.25}{3} + \frac{0.14}{1} + \frac{0.1067}{1} \] Thus, the probability that the seed will germinate is approximately: \[ P(\text{Germinate}) \approx 0.267 \]