If E and F are two events associated with a random experiment, then the probability of combined occurence of events E and F whereas event F has al¬ready occured, is multiplication of their probability. i.e., P(E ∩ F) = F(E).P \((\frac{F}{E})\)
Let, for any experiment sample space is S and its two events are E and F.
Let event E has happened and P(E) ≠ 0.
Since E ⊂ S and event E has happen so all elements of S cannot occur, only those elements of S occur which are exist in E. Thus, in this case reduced sample space will be E. Again, all elements of event E cannot occur, but elements of F which exit in E can occur. Set of these elements is F ∩ E or (E ∩ F).
Thus, probability of event F when event E has already happened p \((\frac{F}{E})\) means to find probability of E ∩ F whereas sample space is S.
Similarly, probability of event E when event F has already occurred is given by following formula :
