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Mutually exclusive events are a fundamental concept in probability, which helps in understanding whether two events can happen at the same time or not. An event is a specific outcome that you are interested in when observing a situation, like rolling a dice. Two events are called mutually exclusive when the occurrence of one event rules out the occurrence of the other.

For example, when rolling two dice, a sum of 7 and a sum of 11 cannot be achieved at the same time. This is simply because each roll will result in only one pair of numbers; hence, one sum. Therefore, the events "sum of 7" and "sum of 11" in dice rolling are mutually exclusive events. This means you roll the dice and it sums to either 7 or 11, but not both at once.

It’s like having two doors: you can choose one door to open at a time. If two events are mutually exclusive, you simply add their probabilities to find the probability of either event occurring.

• If Event A and Event B are mutually exclusive, then the probability of either A or B occurring is: \( P(A \text{ or } B) = P(A) + P(B) \).

Understanding the probability of dice rolls involves knowing how likely certain outcomes are when rolling a fair die. A fair die has an equal chance for each face, meaning each number from 1 to 6 is equally likely to land face up.

When rolling two dice, you are dealing with more outcomes, as each die has 6 sides. The total possible results come from every combination of the numbers on the first die with the numbers on the second die, resulting in a total of 36 outcomes. This is because you have 6 options from the green die and 6 from the red die, leading to: \( 6 \times 6 = 36 \).

For specific sums, such as 7 or 11, you look at all the different pairs of numbers on the dice that add up to the target sum. For instance, the pairs \((3,4)\) and \((5,2)\) both add up to a sum of 7. These pairs are considered favorable outcomes. The probability of a sum occurring is given by dividing the number of favorable outcomes by the total number of possible outcomes (36 in this case).

• Probability of a specific sum: \( P(\text{sum}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).

The total number of outcomes is an essential factor in probability as it refers to all possible results of an experiment. To calculate the probability of an event, knowing the total number of outcomes is crucial.

When discussing dice rolls, the calculation involves multiplying the number of sides each die has. Since each die has 6 sides, rolling two dice involves multiplying the outcomes from one die with the other. Hence, you have \( 6 \times 6 = 36 \) possible outcomes.

This approach is based on the fundamental principle of counting, which helps in determining the sample space: the complete set of possible outcomes. Once you know the sample space, you can then find out the probability for any given acceptable outcome by comparing the number of ways this outcome can occur to the total number of outcomes.

• For two dice, each having 6 faces, total outcomes are calculated as: \(6 \times 6\).