The problem asks for the probability that a marble drawn at random from a jar is either red or even-numbered. The jar contains:

• 5 red marbles numbered 1 to 5
• 8 blue marbles numbered 1 to 8

Step 1: Identify Total Number of Marbles

The total number of marbles in the jar is:

• 5 red marbles
• 8 blue marbles

Thus, the total number of marbles is: $5 + 8 = 13$

Step 2: Define the Events

We need to calculate the probability that a marble is either red or even-numbered. This is an "or" probability, and to compute it, we'll use the following approach:

• Let $A$ be the event that the marble is red.
• Let $B$ be the event that the marble is even-numbered.

We are looking for $P(A \cup B)$, which is the probability of either event happening. This is given by the formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Where:

• $P(A)$ is the probability the marble is red,
• $P(B)$ is the probability the marble is even-numbered,
• $P(A \cap B)$ is the probability the marble is both red and even-numbered (i.e., in both categories).

Step 3: Find $P(A)$

The event $A$ (red marble) involves 5 red marbles. So: $P(A) = \frac{5}{13}$

Step 4: Find $P(B)$

The event $B$ (even-numbered marble) includes all even-numbered marbles. These are:

• Red: 2, 4
• Blue: 2, 4, 6, 8

There are 6 even-numbered marbles in total. So: $P(B) = \frac{6}{13}$

Step 5: Find $P(A \cap B)$

The event $A \cap B$ (red and even-numbered) involves red marbles that are also even-numbered. These are red marbles numbered 2 and 4. So there are 2 such marbles: $P(A \cap B) = \frac{2}{13}$

Step 6: Calculate $P(A \cup B)$

Using the formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{5}{13} + \frac{6}{13} - \frac{2}{13} = \frac{9}{13}$

Final Answer

The probability that the marble is either red or even-numbered is: $\boxed{\frac{9}{13}}$

Would you like further details or clarifications?

Here are five related questions you might consider:

1. What is the probability of drawing a blue marble that is odd-numbered?
2. How does the probability change if two marbles are drawn without replacement?
3. What is the probability of drawing a red marble or an odd-numbered marble?
4. If another red marble is added, how would that affect the probability of drawing a red or even-numbered marble?
5. Can this problem be generalized for any number of red and blue marbles?

Tip: When dealing with "or" probability, always consider overlapping events to avoid double-counting in your calculations.