In a garden, 40% of the flowers are roses and the rest are carnations. If 25% of the roses and 10% of the carnations are red, find the probability that a red flower selected at random is a rose.

This problem requires the application of **Bayes' Theorem** to find the probability that a randomly selected red flower is a rose. We are given:

- 40% of the flowers are roses, meaning \( P(\text{Rose}) = 0.40 \) and the rest are carnations, so \( P(\text{Carnation}) = 0.60 \).
- 25% of the roses are red, meaning \( P(\text{Red} \mid \text{Rose}) = 0.25 \).
- 10% of the carnations are red, meaning \( P(\text{Red} \mid \text{Carnation}) = 0.10 \).

### Step 1: Calculate the total probability of selecting a red flower.
We use the **Law of Total Probability** to compute \( P(\text{Red}) \), the probability that a flower is red:

\[
P(\text{Red}) = P(\text{Red} \mid \text{Rose}) P(\text{Rose}) + P(\text{Red} \mid \text{Carnation}) P(\text{Carnation})
\]

Substitute the values:

\[
P(\text{Red}) = (0.25 \times 0.40) + (0.10 \times 0.60)
\]
\[
P(\text{Red}) = 0.10 + 0.06 = 0.16
\]

### Step 2: Use Bayes' Theorem to find the probability that a red flower is a rose.
Bayes' Theorem is given by:

\[
P(\text{Rose} \mid \text{Red}) = \frac{P(\text{Red} \mid \text{Rose}) P(\text{Rose})}{P(\text{Red})}
\]

Substitute the known values:

\[
P(\text{Rose} \mid \text{Red}) = \frac{0.25 \times 0.40}{0.16}
\]
\[
P(\text{Rose} \mid \text{Red}) = \frac{0.10}{0.16} = 0.625
\]

Thus, the probability that a randomly selected red flower is a rose is **0.625** or **62.5%**.

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Would you like further details or have any questions?

### Related Questions:
1. How would the problem change if 50% of the flowers were roses?
2. What is the probability that a flower selected at random is red?
3. How does Bayes' Theorem work in general cases?
4. What happens if 100% of the roses were red?
5. How would you modify the problem if there were a third type of flower in the garden?

### Tip:
Bayes' Theorem is a powerful tool for updating probabilities based on new evidence, and it's widely used in areas like statistics, machine learning, and decision-making.

Learn how to apply Bayes' Theorem to solve a conditional probability problem. Find the probability that a red flower in a garden is a rose, using key concepts like conditional probability and the law of total probability. This problem is suitable for students in Grades 9-12.

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