Question

If \[P\left( A \right) = \dfrac{1}{{12}}\], \[P\left( B \right) = \dfrac{5}{{12}}\], and \[P\left( {B|A} \right) = \dfrac{1}{{15}}\], what is the value of \[P\left( {A \cup B} \right)\]?
A. \[\dfrac{{89}}{{180}}\]
B. \[\dfrac{{90}}{{180}}\]
C. \[\dfrac{{91}}{{180}}\]
D. \[\dfrac{{92}}{{180}}\]

Answer 1

Hint: First we will apply the conditional probability that is \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\] to calculate the value of \[P\left( {A \cap B} \right)\]. Then we will substitute the value of \[P\left( A \right) = \dfrac{1}{{12}}\], \[P\left( B \right) = \dfrac{5}{{12}}\]and \[P\left( {A \cap B} \right)\] in the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\] to calculate \[P\left( {A \cup B} \right)\].

Formula used:
Conditional probability: \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\]
\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]

Complete step by step solution:
Given that, \[P\left( A \right) = \dfrac{1}{{12}}\], \[P\left( B \right) = \dfrac{5}{{12}}\], and \[P\left( {B|A} \right) = \dfrac{1}{{15}}\]
Now we will apply conditional probability \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\].
\[P\left( {B|A} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( A \right)}}\]
Now substitute the value of \[P\left( A \right) = \dfrac{1}{{12}}\] and \[P\left( {B|A} \right) = \dfrac{1}{{15}}\]
\[ \Rightarrow \dfrac{1}{{15}} = \dfrac{{P\left( {A \cap B} \right)}}{{\dfrac{1}{{12}}}}\]
Multiply both sides by \[\dfrac{1}{{12}}\].
\[ \Rightarrow \dfrac{1}{{15}} \times \dfrac{1}{{12}} = P\left( {A \cap B} \right)\]
\[ \Rightarrow P\left( {A \cap B} \right) = \dfrac{1}{{180}}\]
Now we will apply the union of two mutually inclusive formula.
\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]
Now substitute \[P\left( {A \cap B} \right) = \dfrac{1}{{180}}\], \[P\left( A \right)\] and \[P\left( B \right)\]
\[ \Rightarrow P\left( {A \cup B} \right) = \dfrac{1}{{12}} + \dfrac{5}{{12}} - \dfrac{1}{{180}}\]
\[ \Rightarrow P\left( {A \cup B} \right) = \dfrac{{15 + 75 - 1}}{{180}}\]
\[ \Rightarrow P\left( {A \cup B} \right) = \dfrac{{89}}{{180}}\]

Hence option A is correct answer.

Note: Many students often confused with the formulas \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\] and \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( A \right)}}\]. The correct formula is \[P\left( {A|B} \right) = \dfrac{{P\left( {A \cap B} \right)}}{{P\left( B \right)}}\].