the sum of the probabilities of all the complementary events of an experiment is
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Answer:
Probability
The Complement Rule
The Complement of an Event
The complement A′ of the event A consists of all elements of the sample space that are not in A.
Determining Complements of an Event
Let us refer back to the experiment of throwing one die. As you know, the sample space of a fair die is S={1,2,3,4,5,6}. If we define the event A as observing an odd number, then A={1,3,5}. The complement of A will be all the elements of the sample space that are not in A. Thus, A′={2,4,6}.
A Venn diagram that illustrates the relationship between A and A′ is shown below:
This leads us to say that the sum of the possible outcomes for event A and the possible outcomes for its complement, A′, is all the possible outcomes in the sample space of the experiment. Therefore, the probabilities of an event and its complement must sum to 1.
The Complement Rule
The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1.
P(A)+P(A′)=1
As you will see in the following examples, it is sometimes easier to calculate the probability of the complement of an event than it is to calculate the probability of the event itself. Once this is done, the probability of the event, P(A), is calculated using the relationship P(A)=1−P(A′)