Question

If $ P\left( A \right) = P\left( B \right) = x $ and $ P\left( {A \cap B} \right) = P\left( {A' \cap B'} \right) = \dfrac{1}{3} $ , then $ x $ is equal to
1) $ \dfrac{1}{2} $
2) $ \dfrac{1}{3} $
3) $ \dfrac{1}{4} $
4) $ \dfrac{1}{6} $

Answer 1

Hint: First we have to find $ P\left( {A \cup B} \right) $ . So, we will use the property that, $ P\left( {A' \cap B'} \right) = P{\left( {A \cup B} \right)^\prime } $ . Then from this we can find easily $ P\left( {A \cup B} \right) $ , as $ P{\left( {A \cup B} \right)^\prime } = 1 - P\left( {A \cup B} \right) $ . Then to find the value of $ x $ , we will use the formula,
 $ P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right) $ .
Substituting the values, we will get our required answer.

Complete step-by-step answer:
Given, $ P\left( A \right) = P\left( B \right) = x $ and $ P\left( {A \cap B} \right) = P\left( {A' \cap B'} \right) = \dfrac{1}{3} $ .
Now, we know, we can write,
 $ P\left( {A' \cap B'} \right) = P{\left( {A \cup B} \right)^\prime } $
So, $ P\left( {A' \cap B'} \right) = P{\left( {A \cup B} \right)^\prime } = \dfrac{1}{3} $
Also, we know, $ P{\left( {A \cup B} \right)^\prime } = 1 - P\left( {A \cup B} \right) $
Now, substituting the values, we get,
 $ \dfrac{1}{3} = 1 - P\left( {A \cup B} \right) $
Subtracting both sides by $ 1 $ , gives us,
 $ \Rightarrow \dfrac{1}{3} - 1 = - P\left( {A \cup B} \right) $
 $ \Rightarrow - \dfrac{2}{3} = - P\left( {A \cup B} \right) $
Multiplying both sides by $ - 1 $ , we get,
 $ \Rightarrow P\left( {A \cup B} \right) = \dfrac{2}{3} $
Now, we know, $ P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right) $ .
So, substituting all the values in the formula we get,
 $ \Rightarrow \dfrac{2}{3} = x + x - \dfrac{1}{3} $
Simplifying, we get,
 $ \Rightarrow \dfrac{2}{3} = 2x - \dfrac{1}{3} $
Now, adding $ \dfrac{1}{3} $ on both sides of the equation, we get,
 $ \Rightarrow \dfrac{2}{3} + \dfrac{1}{3} = 2x $
 $ \Rightarrow 1 = 2x $
Now, dividing both sides by $ 2 $ , we get,
 $ \Rightarrow \dfrac{1}{2} = x $
Changing the sides,
 $ \Rightarrow x = \dfrac{1}{2} $
Therefore, the correct answer is 1.
So, the correct answer is “Option 1”.

Note: The formula finally used is for if two events occurred together simultaneously. If three events would have occurred simultaneously, then the formula to use would have been
 $ P\left( {A \cup B \cup C} \right) = P\left( A \right) + P\left( B \right) + P\left( C \right) - P\left( {A \cap B} \right) - P\left( {B \cap C} \right) - P\left( {A \cap C} \right) + P\left( {A \cap B \cap C} \right) $