Question

The probabilities of occurrence of two events are respectively $0.21$ and $0.49$. The probability that both occur simultaneously is $0.16$. The probability that neither of the two events occurs is then
A. $0.30$
B. $0.46$
C. $0.14$
D. None of these

Answer 1

Hint: In this question, we are to find the probability of the complement of the union of two events. By using set theory (complement of sets) and addition theorem on probability, the required probability is calculated.

Formula used: According to set theory,
$\begin{align}
  & P(\overline{A\cup B})=1-P(A\cup B) \\
 & P(\overline{A\cap B})=1-P(A\cap B) \\
\end{align}$
A probability is the ratio of favorable outcomes of an event to the total number of outcomes. So, the probability lies between $0$ and $1$.
The probability is calculated by,
$P(E)=\dfrac{n(E)}{n(S)}$
$n(E)$ - favourable outcomes and $n(S)$ - sample.
If there are two events in a sample space, then the addition theorem on probability is given by
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
When two events happen independently, the occurrence of one is not impacted by the occurrence of the other.
For the events $A$ and $B$, $P(A\cap B)=P(A)\cdot P(B)$ if they are independent and $P(A\cap B)=\Phi $ if they are mutually exclusive.

Complete step by step solution: Given that there are two events.
Consider the two events as $A$ and $B$.
It is given that,
The probability of occurrence the event $A$ is $P\left( A \right)=0.21$
The probability of occurrence the event $B$ is $P\left( B \right)=0.49$
And the probability that both occurs simultaneously is $P\left( A\cap B \right)=0.16$
Then, the probability that none of the two occurs is $P\left( \overline{A\cup B} \right)$
We have
$P(\overline{A\cup B})=1-P(A\cup B)$
By the addition theorem on probability,
$P(A\cup B)=\left[ P(A)+P(B)-P(A\cap B) \right]$
On substituting,
$\begin{align}
  & P(\overline{A\cup B})=1-P(A\cup B) \\
 & \text{ }=1-\left[ P(A)+P(B)-P(A\cap B) \right] \\
 & \text{ }=1-\left[ 0.21+0.49-0.16 \right] \\
 & \text{ }=1-0.54=0.46 \\
\end{align}$

Thus, Option (B) is correct.

Note: In this question, the addition theorem on probability is applied for finding the required probability. By substituting the appropriate values, the required probability is calculated. Here we may go wrong with the complimented probability. The complement of the events denotes the “odds against” the random experiment.