<s>[INST] How to find the probability of the event that exactly two of the three events occur? My professor posted some practice problems with answers (without explanation) for our midterm and this was one of them: 
Three independent events have respective probabilities, $\frac{1}{3},\frac{2}{5},\frac{1}{4}$
a.) Find the probability of the event that exactly two of the three events occur
b.) Find the probability of the even that at least two of the three events occur. 
Answer: 
a.)  $\frac{1}{3}\frac{2}{5}\frac{3}{4} +  \frac{1}{3}\frac{3}{5}\frac{1}{4}+  \frac{2}{3}\frac{2}{5}\frac{1}{4}$
b.) $1 -(\frac{1}{3}\frac{3}{5}\frac{3}{4} +  \frac{2}{3}\frac{2}{5}\frac{3}{4}+  \frac{2}{3}\frac{3}{5}\frac{1}{4}+  \frac{2}{3}\frac{3}{5}\frac{3}{4})$
Could someone please explain how he derived the answer (referencing any laws, theorems, properties, formulas, etc)  
 [/INST] In a) he's simply adding up the probabilities.  The first term is the probability that the first two events happen and the third one does not, for instance.
In b) he's getting the complement of the event that no more than one event occurs.  The first three terms are the probabilities that exactly one event occurs (similar to a) and the last term is the probability that none of the events occurs.
As far as justification goes, the multiplications are okay because the events are independent.  (Recall that if $A$ and $B$ are independent events, then $A$ and $B^c$ are also independent.)  The additions are justified because all the events are mutually exclusive.  These are two extremely important facts.  Make sure you understand them.
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