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Even numbers are those numbers which are divisible by 2 without any remainder. In the context of dice, each die has numbers from 1 to 6. Among these, the even numbers are 2, 4, and 6. This means each die has exactly three even numbers.

• Even numbers on a die: 2, 4, 6.
• Total even outcomes per die: 3.

When dealing with probability exercises involving dice, identifying which numbers are even is crucial for calculating specific outcomes needed for your answer. On a single die, since there are six possible outcomes (numbers 1 to 6), and half of them are even, you often deal with situations where finding these specific outcomes forms the basis of further calculations. Knowing which numbers are even helps in determining probabilities where only even numbers are considered.

Total outcomes refer to all possible results that can occur from a random process. When calculating probabilities, knowing the total number of possible results is essential because it forms the denominator of probability fractions. In our dice example, we deal with two independent dice tosses, each having 6 possible outcomes.
To find the total possible pairs of outcomes for tossing two dice:

• Each die has 6 outcomes.
• The pair of dice combined gives us a product of their individual outcomes:
  Thus, \( 6 \times 6 = 36 \)

This means there are 36 possible ways the dice can show numbers. Understanding total outcomes helps in framing the problem correctly and ensures calculations are done using the whole set of possibilities.

Independent events are events whose outcomes do not affect each other. When you toss two dice, the outcome on one die doesn't impact what happens with the other. Each die rolls independently, creating a situation where the probability of one event does not change or influence another.
Key Characteristics of Independent Events:

• Outcome of one does not alter another: tossing a 4 on the first die doesn't change what you might roll on the second.
• Total probability is the product of individual probabilities.

The principle of independence allows us to use multiplication to find total outcomes and probabilities. For example, since each die is independent with 6 outcomes, their combination's total outcome is calculated as:\( 6 \times 6 = 36 \).This independence also applies when we specify constraints, like both dice showing even numbers, ensuring it's purely a function of each die's probability.