Let's break down the problem step by step.

Given Information:

• Proportions of seed types:
  The seeds are mixed in the ratio of 4:4:2 for seeds A1, A2, and A3 respectively.

  • Proportion of A1 = $\frac{4}{4+4+2} = \frac{4}{10} = 0.4$
  • Proportion of A2 = $\frac{4}{10} = 0.4$
  • Proportion of A3 = $\frac{2}{10} = 0.2$

• Germination rates:

  • Germination rate of A1 = 45% or 0.45
  • Germination rate of A2 = 60% or 0.60
  • Germination rate of A3 = 35% or 0.35

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Part (a): Calculate the probability that a randomly chosen seed will germinate.

To calculate this, we need to use the Law of Total Probability. The probability of germination, $P(\text{Germinate})$, is the sum of the probabilities of germination for each seed type, weighted by the proportions of each seed type.

$P(\text{Germinate}) = P(\text{A1}) \times P(\text{Germinate} | \text{A1}) + P(\text{A2}) \times P(\text{Germinate} | \text{A2}) + P(\text{A3}) \times P(\text{Germinate} | \text{A3})$

Substituting the values:

$P(\text{Germinate}) = (0.4 \times 0.45) + (0.4 \times 0.60) + (0.2 \times 0.35)$

Now let's compute this:

$P(\text{Germinate}) = 0.18 + 0.24 + 0.07 = 0.49$

So, the probability that a randomly chosen seed will germinate is 0.49 or 49%.

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Part (b): Calculate the probability that the seed is of type A2, given that a randomly chosen seed germinates.

We need to calculate the conditional probability $P(\text{A2} | \text{Germinate})$, which is given by Bayes' Theorem:

$P(\text{A2} | \text{Germinate}) = \frac{P(\text{A2}) \times P(\text{Germinate} | \text{A2})}{P(\text{Germinate})}$

We already know:

• $P(\text{A2}) = 0.4$
• $P(\text{Germinate} | \text{A2}) = 0.60$
• $P(\text{Germinate}) = 0.49$

Substitute these values into Bayes' Theorem:

$P(\text{A2} | \text{Germinate}) = \frac{(0.4 \times 0.60)}{0.49}$

Now let's compute this:

$P(\text{A2} | \text{Germinate}) = \frac{0.24}{0.49} \approx 0.4898$

So, the probability that the seed is of type A2, given that it germinates, is approximately 0.4898 or 48.98%.

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Summary of Answers:

• (a) The probability that a randomly chosen seed will germinate is 0.49 (or 49%).
• (b) The probability that the seed is of type A2, given that it germinates, is approximately 0.4898 (or 48.98%).

Would you like any further details or clarification? Here are a few related questions to explore:

1. What would the probability be if the germination rates were all the same for each seed type?
2. How would the probability of germination change if the ratio of seed types changed (e.g., 5:3:2)?
3. What happens if the germination rate for A3 were increased to 50%?
4. How would the probability of a seed being of type A1, given that it germinates, be calculated?
5. How can the probabilities be adjusted if the germination rates are not independent (e.g., if they affect each other)?

Tip: Understanding the Law of Total Probability and Bayes' Theorem is key for solving problems involving conditional probabilities and mixtures of events.