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%.

---

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.