SOLUTION: I am learning about the probability of independent events. I just need to know why do we multiply the probabilities. I want to be able to explain the reason if I am asked.

There are exceptions to the rules 
"AND means multiply probabilities" 
and 
"OR means add probabilities".  

What you asked for is WHY this always works:

P(A and B) = P(A)∙P(B [when we know that A is the case]) =

P(A and B) = P(B)∙P(A [when we know that B is the case]) 

If B is the case a third of time, and A is the case half of the time (that is,
when we know that B is the case), then half of those times when B is the case, A
will also be the case. So they both are the case simultaneously half of a third
of the time, so we multiply their probabilities and get (1/3)(1/2) = 1/6, so
they both will be the case AT THE SAME TIME 1/6th of the time. 

Caution: When A rules B out, we may NOT multiply their individual probabilities,
for the probability that they are BOTH the case is ZERO, because they cannot
both happen at the same time. 

Now we need to know the three kinds of pairs of events that are talked about in
probability studies.  We must learn and understand and be able to distinguish
them:

1. A pair of (mutually) INDEPENDENT events
2. A pair of (mutually) DEPENDENT events
3. A pair of MUTUALLY EXCLUSIVE events.

[The word "mutually" is seldom used in the first two cases but is always used
in the third case.]

If a pair of events are such that one event DOES NOT INCREASE or DECREASE the
probability of the other event, then the events are INDEPENDENT.

If a pair of events are such that one event INCREASES or DECREASES the
probability of the other event, then the events are DEPENDENT.

A special case of a pair of DEPENDENT events is the case when one event
DECREASES the probability of the other event all the way to ZERO. In other words
one event COMPLETELY RULES THE OTHER EVENT OUT!  Then the pair of events are
MUTUALLY EXCLUSIVE. 

[For this last case, think of the less common use of "EXCLUSIVE" as 'EXCLUDING'.
One event EXCLUDES the other] 

-------------------

The two formulas that ALWAYS work in ALL three cases are

(1)   P(A or B) = P(A) + P(B) - P(A and B)

(2)   P(A and B) = P(A)∙P(B|A) = P(B)∙P(A|B), where the "|" means "given".

In the case of a pair of INDEPENDENT events, formula (2) above can be
simplified.

In the case of a pair of MUTUALLY EXCLUSIVE events, formula (1) above can be
simplified: 

In cases of pairs of INDEPENDENT events A, B,

P(A given B) = P(A|B) = P(A)

and 

P(B given A) = P(B|A) = P(B)

so (2) above simplifies to 

(2a)   P(A and B) = P(A)∙P(B) in that case.


In the case of pairs of MUTUALLY EXCLUSIVE events,

P(A and B) = 0, so (1) simplifies in that case to

(1a)   P(A or B) = P(A) + P(B)

Caution: Don't get any of the three cases mixed up!  And, I repeat, remember
that there are exceptions to the rules "AND means multiply probabilities" and
"OR means add probabilities".  

Edwin