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).

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).