The probabilities of the events A and AB are known. Find the…
The probabilities of the events $A$ and $AB$ are known. Find the probability of the event $A\overline B$.
Explanation of Method
Split $A$ according to whether $B$ occurs: $A=AB+A\overline B$. The two parts are mutually exclusive (one requires $B$, the other excludes it), so their probabilities add up to $P(A)$. Solving for $P(A\overline B)$ gives the answer.
Step 1: Split $A$
Every outcome of $A$ either belongs to $B$ or not:
\[ A=AB+A\overline B, \qquad (AB)(A\overline B)=A\,B\overline B=\varnothing. \]
Step 2: Addition rule
\[ P(A)=P(AB)+P(A\overline B). \]
Step 3: The answer
\[ \boxed{P(A\overline B)=P(A)-P(AB).} \]
In words: "$A$ but not $B$" has the probability of $A$ minus the probability of the part of $A$ that lies in $B$.
Step 4: Example
Roll a die: $A$ = "even" $=\{2,4,6\}$, $B$ = "at least $4$". $P(A)=\frac12$, $P(AB)=P(\{4,6\})=\frac13$, so $P(A\overline B)=\frac12-\frac13=\frac16$ — indeed $A\overline B=\{2\}$ ✓.
Result
| Answer | |
|---|---|
| $P(A\overline B)$ | $P(A)-P(AB)$ |