Question

If $P\left( A \right) = \dfrac{1}{2}$, $P\left( B \right) = \dfrac{1}{3}$, and $P\left( {A \cap B} \right) = \dfrac{7}{{12}}$, then the value of $P\left( {{A^\prime} \cap {B^\prime}} \right)$ is
A. $\dfrac{7}{{12}}$
B. $\dfrac{3}{4}$
C. $\dfrac{1}{4}$
D. $\dfrac{1}{6}$

Answer 1

Hint: The intersection of sets for two given sets is the set that contains all the elements that are common to both sets. In this question, we first use the direct formula $P\left( {{A^\prime} \cap {B^\prime}} \right)$ and then use the formula for the union of two sets.

Formula Used:
1.$P\left( {{A^\prime} \cap {B^\prime}} \right) = 1 - P\left( {A \cup B} \right)$
2. $P\left( {A' \cup B'} \right) = 1 - \left[ {P(A) + P(B) - P(A \cap B)} \right]$

Complete step by step solution:
We are given that $P\left( A \right) = \dfrac{1}{2}$
$P\left( B \right) = \dfrac{1}{3}$
$P(A \cap B) = \dfrac{7}{{12}}$
In this question, we need to find the value of $P\left( {{A^\prime} \cap {B^\prime}} \right)$
Now, we use the formula of $P\left( {{A^\prime} \cap {B^\prime}} \right)$which is $P\left( {{A^\prime} \cap {B^\prime}} \right) = 1 - P\left( {A \cup B} \right)...\left( 1 \right)$
Now, we know that$P\left( {A \cup B} \right)$can be written as $P\left( {A \cup B} \right) = P(A) + P(B) - P(A \cap B)$
So, by substituting the value of $P\left( {A \cup B} \right)$ in equation (1), we get
 $P\left( {{A^\prime} \cap {B^\prime}} \right) = 1 - \left[ {P(A) + P(B) - P(A \cap B)} \right]...\left( 2 \right)$
Now, by substituting all the given values in equation (2), we get
$P\left( {{A^\prime} \cap {B^\prime}} \right) = 1 - \left[ {\dfrac{1}{2} + \dfrac{1}{3} - \dfrac{7}{{12}}} \right] \\ \Rightarrow 1 - \left[ {\dfrac{{6 + 4 - 7}}{{12}}} \right] \\ \Rightarrow 1 - \left[ {\dfrac{3}{{12}}} \right] \\ \Rightarrow \dfrac{{12 - 3}}{{12}}$
Further solving, we get
$P\left( {{A^\prime} \cap {B^\prime}} \right) = \dfrac{9}{{12}} \\ \Rightarrow \dfrac{3}{4}$

Option ‘B’ is correct

Note: In probability, always follow a step by step procedure for avoiding complications. $\left( {A \cap B} \right)$ is the set of all the elements that are common to both sets $A$ and $B$. An alternative approach to this problem could be by using a Venn diagram which will always help you visualize the problem better.