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.


Here is a formal definition of the events based on the outcomes of throwing a red and green die, each having values from 1 to 6:

### Let the outcomes of the red and green dice be \( r \) and \( g \), respectively.

1. **Event \( E_1 \)**: The number on the green die is 4 or less.
   \[
   E_1 = \{ g \leq 4 \}
   \]
   This includes the outcomes where \( g \in \{1, 2, 3, 4\} \).

2. **Event \( E_2 \)**: The green die shows an odd number.
   \[
   E_2 = \{ g \in \{1, 3, 5\} \}
   \]
   This includes the outcomes where the green die shows 1, 3, or 5.

3. **Event \( E_3 \)**: The product of the outcomes on the two dice is odd.
   \[
   E_3 = \{ r \times g \text{ is odd} \}
   \]
   For the product to be odd, both dice must show odd numbers. So, \( E_3 \) occurs when \( r \in \{1, 3, 5\} \) and \( g \in \{1, 3, 5\} \).

4. **Event \( E_4 \)**: The absolute difference between the values on the two dice is at least 1.
   \[
   E_4 = \{ |r - g| \geq 1 \}
   \]
   This event includes all outcomes except those where the dice show the same number, i.e., \( r \neq g \).

5. **Event \( E_5 \)**: The number on the red die is at least two higher than the number on the green die.
   \[
   E_5 = \{ r \geq g + 2 \}
   \]
   This event includes outcomes where the red die shows a number at least 2 more than the green die.

6. **Event \( E_6 \)**: Both dice show the same number.
   \[
   E_6 = \{ r = g \}
   \]
   This occurs when both the red and green dice show the same value.

7. **Event \( E_7 \)**: The sum of the outcomes on the two dice is no greater than 5.
   \[
   E_7 = \{ r + g \leq 5 \}
   \]
   This includes outcomes where the sum of the dice is 5 or less.

### Any questions on this or want to see these events mapped out on the possible outcomes grid?

Here are 5 related questions that expand on this information:
1. What is the probability of each event occurring when both dice are thrown?
2. What are the possible outcomes that satisfy event \( E_3 \) (the product of the dice being odd)?
3. How does the total number of possible outcomes affect the probability of \( E_6 \) (both dice showing the same number)?
4. What are the complementary events for \( E_1 \) and \( E_7 \)?
5. How can we determine the intersection of events \( E_2 \) and \( E_3 \)?

**Tip**: When dealing with probabilities involving dice, always remember that each die has 6 independent outcomes, which gives 36 possible outcomes in total when throwing two 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 the probability of various events when throwing a red and a green dice. Learn how to define events like the product of outcomes being odd, the absolute difference between dice values, and conditions for the dice to show specific sums or differences. Suitable for students in Grades 6-8.

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