1. conditional probability proof

Let U = R. Let S be a well defined sample space. Let E and F be events such that E is a subset of F. It is the case that Pr( F|E)=1

2. $\displaystyle A \subseteq B\; \Rightarrow \;A \cap B = A$

Be sure that $\displaystyle E \ne \emptyset$.

3. My professor told me to split up the proof into 2 cases:
1) Pr(E)= 0
2) Pr(E) ≠ 0

Case 2 i have solved but case 1 you get a zero in the denominator since Pr(F|E) = Pr( F∩E) / Pr(E) so this formula is useless because it contradicts the definition of conditional probability .He said to argue that the answer does not have to be DNE and to use the given information the E is a subset of F. Any suggestions on how to attack this ??

Thank you !!!

4. In the text material, what is the exact definition of $\displaystyle P(F|E)?$

5. Pr(F|E) = Pr( F∩E) / Pr(E) such that Pr(E) ≠ 0.

6. Originally Posted by nikie1o2
Pr(F|E) = Pr( F∩E) / Pr(E) such that Pr(E) ≠ 0.
Using that definition how is it possible to consider the case $\displaystyle P(E)=0?$

7. well since E is a subset of F and the empty set is a subset of all sets. Pr( F|E) =1 the empty set occurs in both events. Pr(e)=0 had nothing to do with it...

thank you though!