Question

The probabilities of two events \[A\] and \[B\] are \[0.25\] and \[0.50\] respectively. The probability of both events occur simultaneously is \[0.14\]. Then what is the probability that neither \[A\] nor \[B\] occurs?
A. \[0.39\]
B. \[0.25\]
C. \[0.904\]
D. None of these

Answer 1

Hint: Use the formula of the probability of both events happening together and calculate the probability that the event is \[A\] or \[B\]. Then use the complement rule to reach the required answer.

Formula Used: Complement rule: The sum of the probabilities of an event and its complement is equal to 1.
\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\]

Complete step by step solution:
Given:
\[P\left( A \right) = 0.25\], \[P\left( B \right) = 0.50\] and \[P\left( {A \cap B} \right) = 0.14\]
Let’s calculate the probability that the event is \[A\] or \[B\].
Apply the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\].
Substitute the given values.
\[P\left( {A \cup B} \right) = 0.25 + 0.50 - 0.14\]
\[ \Rightarrow \]\[P\left( {A \cup B} \right) = 0.75 - 0.14\]
\[ \Rightarrow \]\[P\left( {A \cup B} \right) = 0.61\]

Now calculate the probability that neither \[A\] nor \[B\] occurs.
The complement of \[P\left( {A \cup B} \right)\] is the probability that neither \[A\] nor \[B\] occurs.
Apply the complement rule.
\[P\left( {A \cup B} \right) + P{\left( {A \cup B} \right)^\prime } = 1\]
Simplify the above equation.
\[P\left( {A \cup B} \right) + P\left( {{A^\prime } \cap {B^\prime }} \right) = 1\] [Since \[P{\left( {A \cup B} \right)^\prime } = P\left( {{A^\prime } \cap {B^\prime }} \right)\]]
\[ \Rightarrow \]\[P\left( {{A^\prime } \cap {B^\prime }} \right) = 1 - P\left( {A \cup B} \right)\]
Substitute \[P\left( {A \cup B} \right) = 0.61\] in the above equation.
\[P\left( {{A^\prime } \cap {B^\prime }} \right) = 1 - 0.61\]
\[ \Rightarrow \]\[P\left( {{A^\prime } \cap {B^\prime }} \right) = 0.39\]

Hence the correct option is A.

Note: Probability means how likely something is to happen. The range of the probability of an event is \[\left[ {0,1} \right]\].
The complement of an event \[A\] is the set of all outcomes in the sample space that are not included in the outcomes of event \[A\]. It is represented by \[A'\], \[{A^C}\] or \[\overline A \].