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
$P\left( {A \cup B \cup C} \right) = P\left( A \right) + P\left( B \right) + P\left( C \right) - P\left( {A \cap B} \right) - P\left( {B \cap C} \right) - P\left( {A \cap C} \right) + P\left( {A \cap B \cap C} \right)$