3.3. The Complement of an Event http://www.ck12.org
The second part of the problem is to calculate the probability ofAusing the complementary relationship. Recall that
P(A) = 1 −P(A′). So by calculatingP(A′), we can easily calculateP(A)by subtracting it from 1.
P(A′) =P(T T) = 1 / 4
and
P(A) = 1 −P(A′) = 1 − 1 / 4 = 3 / 4.
Obviously, we could have gotten the same result if we had calculated the probability of the event ofAoccurring
directly. The next example, however, will show you that sometimes it is easier to calculate the complementary
relationship to find the answer that we are seeking.
Example:
Here is a new kind of problem. Consider the experiment of tossing a coin ten times. What is the probability that we
will observe at least one head?
Solution:
Before we begin, we can write the event as
A={observe at least one head in ten tosses}What are the simple events of this experiment? As you can imagine, there are many simple events and it would take
a very long time to list them. One simple event may look like this:HT T HT HHT T H,anotherT HT HHHT HT H,
etc. Is there a way to calculate the number of simple events for this experiment? The answer is yes but we will learn
how to do this later in the chapter. For the time being, let us just accept that there are 2^10 =1024 simple events in
this experiment.
To calculate the probability, each time we toss the coin, the chance is the same for heads and tails to occur. We can
therefore say that each simple event, among 1024 events, is equally likely to occur. So
P(any simple event among 1024) =