Question

If $F$ is an event of a sample space $S$, then $P(S|F)=$

Hint:

The sample space \(S\) always contains every event. So once event \(F\) has occurred, the event \(S\) is certainly true, and its conditional probability becomes \(1\).

Answer 1

Step 1: Recall the formula of conditional probability.
The conditional probability of an event $A$ given event $B$ is defined as \[ P(A|B)=\frac{P(A\cap B)}{P(B)} \] provided $P(B)\neq0$.

Step 2: Apply the formula to the given case.
Here \[ A=S \] and \[ B=F \] Thus \[ P(S|F)=\frac{P(S\cap F)}{P(F)} \]
Step 3: Simplify the intersection.
Since $F$ is a subset of the sample space $S$, \[ S\cap F=F \] Thus \[ P(S|F)=\frac{P(F)}{P(F)} \]
Step 4: Simplify the fraction.
\[ P(S|F)=1 \]
Step 5: Conclusion.
Hence the conditional probability that the sample space occurs given event $F$ equals $1$.
Final Answer: $\boxed{1}$