Mutually Exclusive vs Independent Events

Mutually exclusive or independent events must be considered before calculating the probability of two or more events occurring.

Two events "I am here" or "I am not here" are mutually exclusive. If ‘event A’ occurs, then ‘event B’ cannot, or vice-versa. To calculate the probabilities for exclusive events ADD the probabilities.

Two events "I am here" and "You are here" are independent. If ‘event A’, then if ‘event B’. To calculate the probabilities for independent events MULTIPLY the probabilities of two unrelated events.

The throw of a single die, outcomes are equally probable. Each throw is a mutually exclusive event. That is, if a die rolls A then it cannot roll B. The probability of rolling A or B with the first die is the sum of the equal probabilities, 1/6 + 1/6 = 1/3.
Independent events are multiplied. The probability of throwing two dice resulting in A and A is the probability of each roll times itself: (1/6) * (1/6) = 1/36th or 2.77%.

The probability of rolling A and B in either order with one dice is 1/3 x 1/6 = 1/18 (2/36). Add or multiply probabilities based on whether mutually exclusive or independent.

The different numbers obtained by the throw of two dice offer a good introduction to the ideas of probability. Possible outcomes for the total of two dice are not equal. Seven is a possible outcome with every throw of two dice. Two and 12 rolls only 1/36 (A and A) or (B and B). The "odds" of three (A or B) are two times greater 2/36. Rolling a 7 is six times greater 6/36, which raises the idea of distinguishable states. With two dice, the first dice throw does not matter as there remains a chance of rolling 7. If you throw one dice and it rolls a 3 then the next dice roll must be a 4 or vise-versa. Outcomes are exclusive if a result directly affects the other. The order does not matter (i.e. rolling a 3 is twice as likely as rolling a 2 since there are two distinguishable ways to get a 3). The probability of rolling a 7 with two dice is 1/6, rolled 6 of a total of 36 possible outcomes.