Chapter 16 Events and Probability Spaces550
Definition 16.6.1.An event with probability 0 is defined to be independent of every
event (including itself). If PrŒBç¤ 0 , then eventAis independent of eventBiff
PrAjBDPrŒAç: (16.7)In other words,AandBare independent if knowing thatBhappens does not al-
ter the probability thatAhappens, as is the case with flipping two coins on opposite
sides of a room.
Potential Pitfall
Students sometimes get the idea that disjoint events are independent. Theopposite
is true: ifA\B D ;, then knowing thatAhappens means you know thatB
does not happen. So disjoint events areneverindependent—unless one of them has
probability zero.
16.6.1 Alternative Formulation
Sometimes it is useful to express independence in an alternate form which follows
immediately from Definition 16.6.1:
Theorem 16.6.2.Ais independent ofBif and only if
PrŒA\BçDPrŒAçPrŒBç: (16.8)Notice that Theorem 16.6.2 makes apparent the symmetry betweenAbeing in-
dependent ofBandBbeing independent ofA:
Corollary 16.6.3.Ais independent ofBiffBis independent ofA.
16.6.2 Independence Is an Assumption
Generally, independence is something that youassumein modeling a phenomenon.
For example, consider the experiment of flipping two fair coins. LetAbe the event
that the first coin comes up heads, and letBbe the event that the second coin is
heads. If we assume thatAandBare independent, then the probability that both
coins come up heads is:
PrŒA\BçDPrŒAçPrŒBçD1
2
1
2
D
1
4
:
In this example, the assumption of independence is reasonable. The result of one
coin toss should have negligible impact on the outcome of the other coin toss. And
if we were to repeat the experiment many times, we would be likely to haveA\B
about 1/4 of the time.