Question

If \[P\left( E \right) = 0\] then \[P{\text{ }}\left( {not{\text{ }}E} \right)\] is
A. \[1\]
B. \[ - 1\]
C. \[0\]
D. \[\dfrac{1}{2}\]

Answer 1

Hint: The given question is about the happening of the event or not happening of the event. Probability of happening of an event is given and we have to find out the not happening of that event where event can be the outcome that is coming after performing a random or specific experiment.

Complete answer:
 In the given question, we are given that: If \[P\left( E \right) = 0\], then what is the value of \[P{\text{ }}\left( {not{\text{ }}E} \right)\]. In this, \[P\left( E \right) = 0\] , it means \[P\left( E \right)\] implies probability of any event which is given \[0\](zero). Now, we have to find out the probability of not E means probability of not happening of E. For solving this, we should know about sample space. Sample space is nothing but all the possible outcomes of any events and probability of sample space is \[1\] (unity). Then, the probability of happening of that event gives the probability of sample space which is \[1\].
That means\[,{\text{ }}P{\text{ }}\left( {happening} \right)\;P{\text{ }}\left( {not{\text{ }}happening} \right)\;P{\text{ }}\left( {sample{\text{ }}space} \right)\]
Here, E is any event, therefore,
\[P\left( E \right)\;P{\text{ }}\left( {not{\text{ }}E} \right)\;P{\text{ }}\left( {sample{\text{ }}space} \right)\]
Also, we know probability of sample space \[ = 1\]
So, above equation will become
\[P{\text{ }}\left( E \right)\;P{\text{ }}\left( {not{\text{ }}E} \right)\] \[ = 1\]
From this equation, \[P{\text{ }}\left( E \right)\] is given \[0\] and we have to find \[P{\text{ }}\left( {not{\text{ }}E} \right)\] .
Therefore, keeping \[P{\text{ }}\left( {not{\text{ }}E} \right)\] to left hand side and taking \[P{\text{ }}\left( E \right)\] to right hand side, we get
\[P{\text{ }}\left( {not{\text{ }}E} \right)\;\; = 1 - P{\text{ }}\left( E \right)\]
Given that, \[P{\text{ }}\left( E \right) = 0\], we get
\[P{\text{ }}\left( {not{\text{ }}E} \right) = 1 - 0\]
\[ = 1\]

So, \[P{\text{ }}\left( {not{\text{ }}E} \right) = 1\] which means probability of not happening of event E is \[1\] which results that probability of happening of event is zero, that means probability of not happening of event is equal to probability of sample space which is equal to \[1\].

Note: Probability of sample space is \[1\] which can be proved by taking the example of tossing of a coin. If we toss a coin, the possible outcomes are head or tail, and probability of getting head is \[\dfrac{1}{2}\] as well as the probability of getting tail is \[\dfrac{1}{2}\] and in the sample space there are these \[2\] events. So, on adding \[\dfrac{1}{2}\] and \[\dfrac{1}{2}\], we get \[1\]. So, probability of sample space is \[1\].