A red dice and a green dice are thrown. Define the following events:

E
1
=
 The number on the green dice is 4 or less.
E
2
=
 The green dice shows an odd number
E
3
=
 The product of the outcomes on the two dice is odd.
E
4
=
 The absolute difference between the values on the two dice is at least one.
E
5
=
 The number on the red dice is at least two higher than the number on the green dice.
E
6
=
 Both dice show the same number.
E
7
=
 The sum is no greater than 5.

Let’s define each of these events clearly based on the outcomes of a red dice and a green dice, both of which have numbers from 1 to 6:

### Event Definitions:

1. **E₁ = The number on the green dice is 4 or less.**
   - This event occurs when the value on the green dice is in the set {1, 2, 3, 4}.
   - Possible outcomes for the green dice: {1, 2, 3, 4}.

2. **E₂ = The green dice shows an odd number.**
   - This event occurs when the value on the green dice is in the set {1, 3, 5}.
   - Possible outcomes for the green dice: {1, 3, 5}.

3. **E₃ = The product of the outcomes on the two dice is odd.**
   - A product of two numbers is odd if both numbers are odd.
   - Possible outcomes: The red dice and the green dice must both show odd numbers, so their values must be from the set {1, 3, 5} for both dice.

4. **E₄ = The absolute difference between the values on the two dice is at least one.**
   - This event occurs when the two dice show different numbers (since any difference between two equal numbers is 0, and the difference must be at least 1).
   - All outcomes where the red dice and the green dice show different numbers are included in this event.

5. **E₅ = The number on the red dice is at least two higher than the number on the green dice.**
   - This event occurs when the value on the red dice is at least 2 greater than the value on the green dice, i.e., \( R - G \geq 2 \) where \( R \) is the number on the red dice and \( G \) is the number on the green dice.
   - Possible outcomes: For each green dice value \( G \), the red dice must show a number at least 2 higher than \( G \). For example, if \( G = 1 \), then \( R \geq 3 \); if \( G = 2 \), then \( R \geq 4 \), and so on.

6. **E₆ = Both dice show the same number.**
   - This event occurs when the value on the red dice is equal to the value on the green dice.
   - Possible outcomes: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}.

7. **E₇ = The sum is no greater than 5.**
   - This event occurs when the sum of the numbers on the red and green dice is less than or equal to 5, i.e., \( R + G \leq 5 \).
   - Possible outcomes: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (3,1), (3,2), and (4,1).

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Would you like me to expand on the probability or further explanations for any of these events?

### Related Questions:
1. What is the probability of event \( E_3 \) occurring when two dice are rolled?
2. How can we calculate the probability of event \( E_5 \)?
3. What is the complement of event \( E_1 \)?
4. How many outcomes are included in event \( E_4 \)?
5. What is the probability that event \( E_7 \) occurs?

### Tip:
When calculating probabilities of events involving dice rolls, always remember that there are a total of 36 possible outcomes (6 sides on the red dice × 6 sides on the green dice).

A red dice and a green dice are thrown. Define the following events: E₁ = The number on the green dice is 4 or less. E₂ = The green dice shows an odd number. E₃ = The product of the outcomes on the two dice is odd. E₄ = The absolute difference between the values on the two dice is at least one. E₅ = The number on the red dice is at least two higher than the number on the green dice. E₆ = Both dice show the same number. E₇ = The sum is no greater than 5.

Explore how to define events based on the outcomes of rolling a red and green dice. This problem covers key probability concepts such as calculating outcomes where the product of two dice is odd, the absolute difference between values is at least 1, and the sum of dice is no greater than 5. Suitable for students in Grades 6-8.

Math Problem Statement

E 1

The number on the green dice is 4 or less. E 2

The green dice shows an odd number E 3

The product of the outcomes on the two dice is odd. E 4

The absolute difference between the values on the two dice is at least one. E 5

The number on the red dice is at least two higher than the number on the green dice. E 6

Both dice show the same number. E 7

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