Question

Explain why $P(AorB)\ne P\left(A\right)+P\left(B\right)$. Then use the general addition rule to find$P(AorB)$.

Short answer

The likelihood of $P(AorB)\ne P\left(A\right)+P\left(B\right)$and the occurrence A or B

Step-by-step solution

Step 1. Given Information    

$$\begin{gathered} P(AorB)\ne P(A\cup B) \\ P(AorB)\ne P\left(A\right)+P\left(B\right) \end{gathered}$$

Step 2. Concept Used   

Definition of the probability addition theorem: Assume $A$and $B$are two events in a random experiment, and you want to know whether $A$or $B$has a higher likelihood. Then apply the probability addition theorem.

Step 3. Calculation    

let A and B be two events of a random experiment using the addition theorem of probability. That is$$\begin{gathered} P(AorB)=P(A\cup B) \\ P(AorB)=P\left(A\right)+P\left(B\right)-P(A\cap B) \end{gathered}$$

If A and B are not mutually exclusive events. So, $$\begin{gathered} P(A\cap B)\ne 0 \\ P(AorB)\ne P\left(A\right)+P\left(B\right) \end{gathered}$$

Therefore, The event and the probability of $P(AorB)\ne P\left(A\right)+P\left(B\right)$