Two events, A and B, are said to be independent if
P(A and B both happening)=P(A happening).P(B happening).
The two events do not affect one another.
A more general result can be derived for conditional
probabilities.
Two events, A and B, are said to be conditionally independent if
P(A and B both happening | Another event C) = P(A happening | C).P(B
happening | C).
Where we already have information about the situation (through our
knowledge of event C), knowledge of event A will not enable us to change
our estimate of the probability of event B.
e.g. The probability of me buying a hamburger and a cola, given that
I am already at the snack-bar, may well just equal the chances of my feeling
like eating a burger multiplied by the probability of my feeling like drinking
a cola, since I am not particularly bothered about what I drink when I
eat particular foods. Formally,
P(burger and cola| at snack bar)= P(burger| at snack bar). P(cola |
at snack bar).
It does not follow that events which are conditionally independent
will also be independent.
e.g. I have to walk past other shops before I reach the snack bar.
Some of these shops sell cola but no burgers. If it is a hot day, I may
not feel like walking all the way to the snack bar: I may just use the
first shop I come to. Although I have a probability of buying a cola (depending
on how much I fancy one) and a probability of buying a burger, since my
choice of shop will influence the outcome, and this choice also depends
on how lazy I am feeling, it will not necessarily be true that
P(burger and cola)= P(burger). P(cola).