Either-Or Probabilities

Whatâ€™s the probability of getting either heads or tails when flipping a coin once? Since the only possible outcomes are heads or tails, we expect the probability to be 100%, or 1: .

Note: That the events heads and tails are independent. That is, if heads occurs, then tails cannot (and vice versa).

Whatâ€™s the probability of drawing a red marble or a green marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles? There are 4 red marbles out of 14 total marbles. So the probability of selecting a red marble is 4/14 = 2/7. Similarly, the probability of selecting a green marble is 5/14. So the probability of selecting a red or green marble is
.

Note: Again that the events are independent. For instance, if a red marble is selected, then neither a blue marble nor a green marble is selected

These two examples can be generalized into the following rule for calculating either-or probabilities:

 To calculate either-or probabilities, add the individual probabilities (only if the events are independent).

The probabilities in the two immediately preceding examples can be calculated more naturally by adding up the events that occur and then dividing by the total number of possible events.
For the coin example, we get 2 events (heads or tails) divided by the total number of possible events, 2 (heads and tails): 2/2 = 1.

For the marble example, we get 9 (= 4 + 5) ways the event can occur divided by 14 (= 4 + 5 + 5) possible events: 9/14.
If itâ€™s more natural to calculate the either-or probabilities above by adding up the events that occur and then dividing by the total number of possible events, why did we introduce a second way of calculating the probabilities? Because in some cases, you may have to add the individual probabilities.
For example, you may be given the individual probabilities of two independent events and be asked for the probability that either could occur. You now know to merely add their individual probabilities.