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In the context of probability, two events are considered independent if the occurrence or outcome of one event does not affect the occurrence or outcome of the other.
For example, when you roll two dice, like a green and a red one, the result of the roll of one die does not impact the other.
Each die has an equal chance of landing on any of its six faces, regardless of the result of the other die. This independence means that the probability calculations for one die do not interfere with or change those of the other die.

To determine independence, you can check if the probability of both events happening simultaneously is equal to the product of their individual probabilities. In mathematical terms, this can be expressed as:

• If event A and event B are independent, then \[ P(A \cap B) = P(A) \cdot P(B) \]

For the dice example specifically, the outcome of rolling a number on the green die is independent of any outcome on the red die, making these two events independent.

Mutually exclusive events are events that cannot happen at the same time. If one event occurs, the other cannot.
A typical example would be drawing a single card from a deck and determining if it's a heart or a spade. It can't be both.

Unlike independent events, mutually exclusive scenarios means you don't multiply probabilities, but instead add them. If events A and B are mutually exclusive:

• \( P(A \text{ or } B) = P(A) + P(B) \)

In the dice context, consider rolling a '1' on the green die and '2' on the red die in one roll, and rolling a '2' on the green die and '1' on the red die in another. These two outcomes can't occur simultaneously in a single pair of rolls, thus they are mutually exclusive.
So, when calculating their combined probability, you simply add the probabilities of the individual outcomes.

The probability of combined events refers to calculating the likelihood of either one event or another occurring.
For independent events, if you want to find the probability of either event A occurring or event B occurring, and both events cannot happen simultaneously (mutually exclusive), you'll add the probabilities together.

For example, to find the probability of rolling a '1' on the green die and a '2' on the red die in one roll, or a '2' on the green die and a '1' on the red die in another, you would first determine each event's individual probabilities using their independence:

• The probability of rolling a '1' on the green die and a '2' on the red die is \(\frac{1}{36}\).
• The probability of rolling a '2' on the green die and a '1' on the red die is also \(\frac{1}{36}\).

Because these outcomes can't occur at the same time, they are mutually exclusive. Hence, the probability of either outcome is the sum of the probabilities: \[\frac{1}{36} + \frac{1}{36} = \frac{1}{18}.\]Remember, when calculating the combined probability of mutually inclusive events, the approach differs, as allowance needs to be made for events overlapping.