Question

If A and B are two events such that\[P(A \cup B) = P(A \cap B)\], then the incorrect statement amongst the following statements is:
A) A and B are equally likely
B) \[P(A \cap B') = 0\]
C) \[P(A' \cap B) = 0\]
D) \[P\left( A \right) + P\left( B \right) = 1\]

Answer 1

Hint: Here we will check each of the options whether it is true or false.
We will use the following properties:-
\[P(A \cup B) = P\left( A \right) + P\left( B \right) - P(A \cap B)\]
\[\begin{gathered}
  P\left( {A'} \right) = 1 - P\left( A \right) \\
  P\left( {B'} \right) = 1 - P\left( B \right) \\
\end{gathered} \]

Complete step-by-step answer:

Let us first consider option A: -
It states that:-
A and B are equally likely
Now we know that:
\[P(A \cup B) = P\left( A \right) + P\left( B \right) - P(A \cap B)\]
And it is given that:
\[P(A \cup B) = P(A \cap B)\]
Therefore, substituting the values we get:-
\[P(A \cup B) = P\left( A \right) + P\left( B \right) - P(A \cup B)\]
Solving it further we get:-
\[P\left( A \right) + P\left( B \right) = P(A \cup B) + P(A \cup B)\]
Which implies
\[P\left( A \right) = P(A \cup B)\]and\[P\left( B \right) = P(A \cup B)\]
Therefore, A and B are equally likely
Hence Option A is true.
Now let us consider option B:
It states that:-
\[P(A \cap B') = 0\]
Since A and B are equally likely.
Therefore,
\[P(A \cap B) = P\left( A \right)orP\left( B \right)\]
This implies \[P(A \cap B') = 0\]
Therefore, option B is true.

Now let us consider option C:
It states that:-
\[P(A' \cap B) = 0\]
Since A and B are equally likely.
Therefore,
\[P(A \cap B) = P\left( A \right)orP\left( B \right)\]
This implies \[P(A' \cap B) = 0\]
Therefore, option C is true.
Now let us consider option D:
It states that:
\[P\left( A \right) + P\left( B \right) = 1\]
We know that:-
\[P(A \cup B) = P\left( A \right) + P\left( B \right) - P(A \cap B)\]
And it is given that:
\[P(A \cup B) = P(A \cap B)\]
Therefore, substituting the values we get:-
\[P(A \cup B) = P\left( A \right) + P\left( B \right) - P(A \cup B)\]
Solving it further we get:-
\[P\left( A \right) + P\left( B \right) = 2P(A \cup B)\]
Therefore, option D is incorrect.

Therefore, the answer is option D.

Note: In such types of questions we need to check each of the options whether they are true or false.
Also, students should note that the union of two sets is the set which contains all the elements of both the sets writing the common elements once.
Intersection of two sets is the set containing only the common elements of both the sets.