Let $A$ and $B$ be two Boolean variables. Consider the following Boolean expression $C=AB$. The truth table for $C$ is given as:
| A | B | C |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
If the Boolean variables $A$ and $B$ are sampled assuming independence from an exhaustive random experiment, it can be concluded from the above truth table that $P[A=1]=\frac{2}{4}=0.5$ and $P[B=1]=\frac{2}{4}=0.5$. For computing $P[C=1]$, there can be possibly two methods:
- By observation of number of ones in above truth table for $C$, such that $P[C=1]=\frac{1}{4}=0.25$
- By using the SOP Boolean expression of $C$ from truth table, such that $P[C=1]=P[A=1]*P[B=1]=0.5 *0.5=0.25$
My doubt is that do methods 1 and 2 always yield the same result, even if $P[A=1]\neq 0.5$ or $P[B=1]\neq 0.5$ ?
Any resource material for such analysis would also be helpful.
Regards.