Question

If $\text{A}$ and $\text{B}$ are two events, then the probability of the event that at most one of $\text{A, B}$ occurs, is

• A. $\text{P(A’∩B) +P(A∩B’) + P(A’∩B’)}$
• B. $\text{1-P(A∩B)}$
• C. $\text{P(A’)+P(B’)+P(AUB)-1}$
• D. All of the above

Answer 1

The correct option is D

All of the above

Explanation for the correct option :

At most one of $\text{A}$ and $\text{B}$ occurs means that the two events cannot happen simultaneously.

Step $1$. The probability of at most one of the two events $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$

The probability of at most one of the two events $\text{A}$ and $\text{B}$ occurs is the sum of the probabilities of none of the two events occurs and exactly one of the two events occurs.

Probability of none of the two events occurs $=\text{P}\left(\text{A'∩B'}\right)$

So, option $\text{"A"}$ is correct.

Step $2$. Probability of one of the two events $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ occurs

Probability of one of the two events $\text{A}$ and $\text{B}$ occurs $=\text{P}\left(\text{A∩B}\right)\text{'}$

$=\; 1-\text{P}\left(\text{A∩B}\right)$

So, option $\text{"B"}$ is correct.

Step $3$. Probability of one of the two events $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ occurs

$\text{P}\left(\text{A∩B}\right)\text{' = 1-}\text{P}\left(\text{A∩B}\right)$

$\text{=1−[}\text{P}\text{(A)}\text{+P}\text{(}\text{B)−P(}\text{A∪B}\text{)}\text{]}$ $\left[\mathbf{\text{∵P}}\left(\text{A∩B}\right)\mathbf{\text{=}}\mathbf{\text{P(A)+P(B)−P(A∪B)}}\right]$

Adding and subtracting one in the above expression,
$\text{= 1-P(A)+1-P(B)+P(AUB)-1}$
$=\text{P(A′)+P(B′)+P(A∪B)−1}$

So, option $\mathbf{"}\mathbf{\text{C"}}$ is correct.

Hence, option $\mathbf{"}\mathbf{\text{D"}}$is correct.