### A.P. STATISTICS LESSON 6.3 ( DAY 2 )

```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.
and  and  =   ×    ×    and
 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    is generally not
equal to the unconditional probability P(B).


That is because the occurrence of event A generally
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)
We now see that the multiplication rule for
independent events   and  =   × (), is a
special case of the general multiplication rule,
and  =   × ( )
EXAMPLE 6.25

See example 6.25 on p.376

Homework: p.378-381 #’s 63, 64, 68, 70-73, 75, &
77
```