To solve these problems, we need to use basic probability principles.

Part (a)

Problem: A marble is chosen at random. If you're told the marble is red, what is the probability that it has the number 5 on it?

Solution: Given:

• There are 6 red marbles, numbered 1 to 6.

The event of choosing a red marble with the number 5 on it out of the 6 red marbles is a simple probability calculation.

$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of red marbles}} = \frac{1}{6}$

Part (b)

Problem: The first marble is replaced, and another marble is chosen at random. If you're told the marble has the number 1 on it, what is the probability the marble is red?

Solution: Given:

• There are 18 marbles in total (6 red + 12 blue).
• Only 2 marbles have the number 1 on them (1 red and 1 blue).

We need to find the probability that the marble is red given that it has the number 1 on it. This is a conditional probability problem.

The total number of marbles with the number 1 is 2. Out of these, only 1 is red.

$\text{Probability} = \frac{\text{Number of red marbles with number 1}}{\text{Total number of marbles with number 1}} = \frac{1}{2}$

So, the probabilities are:

• (a) $\frac{1}{6} \approx 0.167$
• (b) $\frac{1}{2} = 0.5$

Would you like more details on any part of the solution, or do you have any other questions?

Here are 5 questions you might want to ask next:

1. How do you calculate the probability of multiple events?
2. What is the probability of drawing two specific marbles in a row without replacement?
3. How does conditional probability change if the total number of marbles is altered?
4. What is Bayes' Theorem and how does it apply to these problems?
5. How can we visualize probability problems using diagrams?

Tip: When solving probability problems, carefully define and count the favorable and possible outcomes for clarity.