Non-Mutually-Exclusive Outcomes

For the addition rule to apply, the events must be mutually exclusive. Now consider the following example.

Example 1

What is the probability of the outcome of at least one head in two coin flips?

Should you add the two probabilities as in the preceding examples? In the first example, you added the probability of getting a head and the probability of getting a tail because those two events were mutually exclusive in one flip. In the second example, the probability of getting a spade was added to the probability of getting a club because those two outcomes were mutually exclusive in one draw. Now when you have two flips, should you add the probability of getting a head on the first flip to the probability of getting a head on the second flip? Are these two events mutually exclusive?

Of course, they are not mutually exclusive. You can get an outcome of a head on one flip and a head on the second flip. So, because they are not mutually exclusive, you cannot use the addition rule. If you did use the addition rule, you would get

  equation

or certainty, which is absurd. There is no certainty of getting at least one head on two flips. (Try it several times, and see that there is a possibility of getting two tails and no heads.)