As promised, we will now start developing generalizations of
the different calculations that we carried out in the
context of the radar example.
The first kind of calculation that we carried out goes under
the name of the multiplication rule.
And it goes as follows.
Our starting point is the definition of conditional
probabilities.
The conditional probability of A given another event, B, is
the probability that both events have occurred divided
by the probability of the conditioning event.
We now take the denominator term and send it to the other
side of this equality to obtain this relation, which we
can interpret as follows.
The probability that two events occur is equal to the
probability that a first event occurs, event B in this case,
times the conditional probability that the second
event, event A, occurs, given that event B has occurred.
Now, out of the two events, A and B, we're of course free to
choose which one we call the first event and which one we
call the second event.
So the probability of the two events happening is also equal
to an expression of this form, the probability that A occurs
times the conditional probability that B occurs,
given that A has occurred.
We used this formula in the context of a tree diagram.
And we used it to calculate the probability of a leaf of
this tree by multiplying the probability of taking this
branch, the probability that A occurs, times the conditional
probability of taking this branch, the probability that
event B also occurs given that event A has occurred.
How do we generalize this calculation?
Consider a situation in which the experiment has an
additional third stage that has to do with another event,
C, that may or may not occur.
For example, if we have arrived here, A and B have
both occurred.
And then C also occurs, then we reach this particular leaf
of the tree.
Or there could be other scenarios.
For example, it could be the case that A did not occur.
Then event B occurred, and finally, event C did not
occur, in which case we end up at this particular leaf.
What is the probability of this scenario happening?
Let us try to do a calculation similar to the one that we
used for the case of two events.
However, we need to deal here with three events.
What should we do?
Well, we look at the intersection of these three
events and think of it as the intersection of a composite
event, A complement intersection B, then
intersected with the event C complement.
Clearly, you can form the intersection of three events
by first taking the intersection of two of them
and then intersecting with a third.
After we group things this way, we're dealing with the
probability of two events happening, this composite
event and this ordinary event.
And the probability of two events happening is equal to
the probability that the first event happens, and then the
probability that the second event happens, given that the
first one has happened.
Can we simplify this even further?
Yes.
The first term is the probability
of two events happening.
So it can be simplified further as the probability
that A complement occurs times the conditional probability
that B occurs, given that A complement has occurred.
And then we carry over the last term
exactly the way it is.
The conclusion is that we can calculate the probability of
this leaf by multiplying the probability of the first
branch times the conditional probability of the second
branch, given that the first branch was taken, and then
finally multiply with the probability of the third
branch, which is the probability that C complement
occurs, given that A complement and B
have already occurred.
In other words, we can calculate the probability of a
leaf by just multiplying the probabilities of the different
branches involved and where we use conditional probabilities
for the intermediate branches.
At this point, you can use your imagination to see that
such a formula should also be valid for the case of more
than three events.
The probability that a bunch of events all occur should be
the probability of the first event times a number of
factors, each corresponding to a branch in a
tree of this kind.
In particular, the probability that events A1, A2, up to An
all occur is going to be the probability that the first
event occurs times a product of conditional probabilities
that the i-th event occurs, given that all of the previous
events have already occurred.
And we obtain a term of this kind for every event, Ai,
after the first one, so this product ranges from 2 up to n.
And this is the most general version of the multiplication
rule and allows you to calculate the probability of
several events happening by multiplying probabilities and
conditional probabilities.