Question

What is the probability of the complement event of impossible events?
${\text{A}}{\text{.}}$ 0.5
${\text{B}}{\text{.}}$ 0
${\text{C}}{\text{.}}$ 1
${\text{D}}{\text{.}}$ 0.46

Answer 1

Hint: Here, we will proceed by letting any event E as any impossible event and then, we will use the property of probability of complement events i.e., the sum of the probabilities of the event and its complement is always equal to 1 [P(E) + P(E’) = 1].

Complete Step-by-Step solution:
Let us consider any event E which is an impossible event.
As we know that the probability of an impossible event is always zero because that event can never occur.
i.e., P(E) = 0$ \to (1)$
Complement of any event is the event that is exactly opposite to that event. So, the complement of impossible events are the possible events. Here, the complement of event E is denoted by event E’.
Also, the sum of the probability of any event and the probability of the complement of that event is always equal to 1.
i.e., For any event E, P(E) + P(E’) = 1$ \to (2)$ which means the sum of the probability of impossible events and the probability of possible events is always equal to 1.
By substituting equation (1) in equation (2), we get
0 + P(E’) = 1
$ \Rightarrow $P(E’) = 1
Therefore, the probability of the complement event of impossible events is always equal to 1.
Hence, option C is correct.

Note: In probability theory, the complement of any event A is the event [not A or A’] i.e., the event that A does not occur. The event A and its complement [not A or A’] are mutually exclusive and exhaustive events where mutually exclusive events means that there is nothing common between these events and mutually exhaustive events means that these events can never be both true at the same time.