Comprehension

An NGO organises a charity event in which they decide to distribute woollen caps to protect children from winter. The caps to be distributed are in three separate boxes, Box I has 30 red caps, Box II has 20 red and 10 green caps, and Box III has 30 green caps. The probability that a Box i is selected and a cap picked out is i/6, where i = 1, 2, 3.  
Based on the above information, answer the following questions :

Question: 1

What is the probability that he selects a red cap?

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Always simplify calculations by working with a common denominator from the start. Here, keeping everything in terms of eighteenths makes addition painless: \[ P(R) = \frac{3}{18} + \frac{4}{18} + 0 = \frac{7}{18} \]
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Question: 2

If he selects a green cap, what is the probability that the cap has come from Box II?

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When applying Bayes' theorem, keep the denominators of individual components unsimplified (e.g., keeping \(\frac{2}{18}\) instead of simplifying to \(\frac{1}{9}\)). This guarantees they will directly cancel out with the denominator of the total probability, saving precious calculation time!
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Solution and Explanation

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