CHAPTER 6
Section 6.3 Part 2 – General Probability Rules
EXTENDED MULTIPLICATION RULES
 Recall
that the union of a collection of events is
the event that any of them occur.
 The
intersection of any collection of events is the
event that all of the events occur.
 To
extend the multiplication rule to the
probability that all of several events occur, the key
is to condition each event on the occurrence of all
of the preceding events.
𝑃 𝐴 and 𝐵 and 𝐶 = 𝑃 𝐴 × 𝑃 𝐵 𝐴 × 𝑃 𝐶 𝐴 and 𝐵
 See
example 6.22 on p.372
TREE DIAGRAMS REVISITED
 Probability
problems often require us to
combine several of the basic rules into a more
elaborate calculation.
 Each
segment in the tree is one stage of the
problem. Each branch shows a path that must
be taken to achieve the next branch.
 The
probability written on each segment is the
conditional probability that that segment is
given after reaching that point from each
branch.
EXAMPLE 6.23

See example 6.23 on p.373
TREE DIAGRAMS CONT.

The tree diagrams combine the addition and
multiplication rules:

The multiplication rule says that the probability of
reaching the end of any complete branch is the
product of the probabilities written on its segments.

The probability of any outcome is then found by
adding the probabilities of all branches that are part
of that event.
INDEPENDENCE

The conditional probability 𝑃 𝐵 𝐴 is generally not
equal to the unconditional probability P(B).


That is because the occurrence of event A generally
gives us some additional information about whether or
not event B occurs.
If knowing that A occurs gives no additional
information about B, then A and B are independent
events.
INDEPENDENT EVENTS

The formal definition states:


Two events A and B that both have positive
probability are independent if:
𝑃 𝐵 𝐴 = P(B)
We now see that the multiplication rule for
independent events 𝑃 𝐴 and 𝐵 = 𝑃 𝐴 × 𝑃(𝐵), is a
special case of the general multiplication rule,
𝑃 𝐴 and 𝐵 = 𝑃 𝐴 × 𝑃(𝐵 𝐴)
EXAMPLE 6.25

See example 6.25 on p.376

Homework: p.378-381 #’s 63, 64, 68, 70-73, 75, &
77