Let's solve this problem based on the details provided in the image.

Problem Breakdown:

• 4 red marbles numbered $1, 2, 3, 4$.
• 6 blue marbles numbered $1, 2, 3, 4, 5, 6$.
• A marble is drawn at random, so there are $4 + 6 = 10$ marbles in total.

We will compute the probabilities for each event.

(a) The marble is red:

The number of red marbles is $4$, and the total number of marbles is $10$.

$P(\text{red marble}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{4}{10} = 0.4$

(c) The marble is red or odd-numbered:

• Red marbles: $1, 2, 3, 4$ (all 4 are red).
• Odd-numbered marbles: $1, 3, 5$ (appear in both red and blue sets).
  • Odd-numbered red marbles: $1, 3$ (2 marbles).
  • Odd-numbered blue marbles: $1, 3, 5$ (3 marbles).

Using the formula for union of two events: $P(\text{red or odd}) = P(\text{red}) + P(\text{odd}) - P(\text{red and odd})$

• $P(\text{red}) = \frac{4}{10}$,
• $P(\text{odd}) = \frac{5}{10}$,
• $P(\text{red and odd}) = \frac{2}{10}$ (only $1, 3$ are both red and odd).

$P(\text{red or odd}) = \frac{4}{10} + \frac{5}{10} - \frac{2}{10} = \frac{7}{10} = 0.7$

(d) The marble is blue or even-numbered:

• Blue marbles: $1, 2, 3, 4, 5, 6$ (all 6 are blue).
• Even-numbered marbles: $2, 4, 6$ (appear in both red and blue sets).
  • Even-numbered blue marbles: $2, 4, 6$ (3 marbles).
  • Even-numbered red marbles: $2, 4$ (2 marbles).

Using the formula for union of two events: $P(\text{blue or even}) = P(\text{blue}) + P(\text{even}) - P(\text{blue and even})$

• $P(\text{blue}) = \frac{6}{10}$,
• $P(\text{even}) = \frac{5}{10}$,
• $P(\text{blue and even}) = \frac{3}{10}$ (only $2, 4, 6$ are both blue and even).

$P(\text{blue or even}) = \frac{6}{10} + \frac{5}{10} - \frac{3}{10} = \frac{8}{10} = 0.8$

Final Answers:

• (a) 0.4
• (c) 0.7
• (d) 0.8

Would you like a step-by-step walkthrough of the formulas used here or further explanations?

5 Related Questions:

1. What is the probability of drawing a blue marble?
2. What is the probability of drawing an even-numbered red marble?
3. How would the probabilities change if the number of red and blue marbles were equal?
4. Can you compute the probability of drawing a red and even-numbered marble?
5. What is the probability of not drawing a red marble?

One Tip:

Always ensure the probabilities of overlapping events (e.g., red and odd-numbered) are subtracted when calculating combined probabilities (using the union formula).