Question

If A and B are two complementary events then what is the relation between $P\left( A \right)$ and $P\left( B \right)$ ?

Answer 1

Hint: Assume an event of tossing a fair coin where event A is getting a head as an outcome and Event B is getting a tail as another outcome. Since event A and event b cannot occur at the same time so, event A and b are complementary to each other. Now, find the probability of getting a head And tail. Then, conclude the relationship between the probability of event A and event B.

Complete step-by-step solution -
According to the question, we have two events that are event A and event B. It is given that event A and event B are complementary to each other.
We know that when two events are complementary to each other then, both events cannot occur at the same time.
We also know that when two complementary events are taken together, it includes all the possible outcomes for the whole event.
Let us understand with an example.
For instance, assume an event of tossing a fair coin where event A is getting a head as an outcome and Event B is getting a tail as another outcome.
In a fair coin, we have only two possible outcomes. One is getting a head and the other is getting a tail. Here, we cannot get the head and tail at the same time. Also, event A and event B includes all the possible outcomes for the whole event. In this whole event,
The probability of getting a head, $P\left( A \right)$ = $\dfrac{1}{2}$ …………………….(1)
The probability of getting a head, $P\left( B \right)$ = $\dfrac{1}{2}$ …………………….(2)
Now, adding equation (1) and equation (2), we get
$P\left( A \right)+P\left( B \right)=\dfrac{1}{2}+\dfrac{1}{2}$
$\Rightarrow P\left( A \right)+P\left( B \right)=1$ ……………………………..(3)
From equation (3), we have got the relation between $P\left( A \right)$ and $P\left( B \right)$ .
When two events are complementary then the sum of their probabilities is equal to 1.

Note: We can also solve this question directly. We know that when two complementary events are taken together, it includes all the possible outcomes for the whole event, and for the whole event, the probability of the whole event is equal to 1. When event A and event B are combined then we get the whole event and the probability of the whole event is equal to 1. Therefore, the sum of the probabilities of event A and event B is equal to 1.