Transcript Section 6.3 - Windsor High School

AP
STATISTICS
SECTION 6.3
General Probability Rules
WHAT WE ALREADY KNOW
WHAT WE ALREADY KNOW CONT…
EVENTS THAT ARE NOT DISJOINT
EXAMPLE – PROSPERITY AND EDUCATION
Call a household prosperous if its income exceeds
$100,000. Call the household educated if the
householder completed college. Select an
American household at random, and let A be the
event that the selected household is prosperous
and B the event that it is educated. According to
the Census Bureau, P(A) = 0.134, P(B) = 0.254,
and the joint probability that a household is both
prosperous and educated is P(A and B) = 0.080.
 What is the probability P(A or B) that the
household selected is either prosperous or
educated?
 P(A or B) = P(A) + P(B) – P(A and B) = 0.134 +
0.254 – 0.080 = 0.308

VENN DIAGRAM – PROSPERITY AND
EDUCATION
Draw a Venn diagram that shows the relation
between events A and B. Indicate each of the
events listed below on your diagram and use the
information from the previous problem to
calculate the probability of each event. Finally,
describe in words what each event is.
 A and B
 A and Bc
 Ac and B
 Ac and Bc

SOLUTION

Venn Diagram
SOLUTION CONT…
Household is both prosperous and educated;
P(A and B) = 0.080
 Household is prosperous but not educated;
P(A and Bc) = P(A) – P(A and B)
= 0.134 – 0.080 = 0.054
 Household is not prosperous but is educated;
P(Ac and B) = P(B) – P(A and B)
= 0.254 – 0.080 = 0.174
 Household is neither prosperous nor educated;
P(Ac and Bc) = 1 – 0.308 = 0.692

CONDITIONAL PROBABILITY



The probability we assign to an
event can change if we know that
some other event has occurred.
Slim is a professional poker
player. He is dealt a hand of four
cards. Find the probability that
he gets an ace.
P(ace) =
Find the conditional probability
that slim gets another ace given
he has an ace already in his
hand.
P(ace | 1 ace in 4 visible cards) =
CONDITIONAL PROBABILITY CONT…
The notation P(B|A) is a conditional probability
and means the probability of B given that A has
occurred.
EXAMPLE – MUNICIPAL WASTE
Municipal Waste Collected in the U.S. (in millions of tons)
Paper
Aluminum
Glass
Plastic
Other
Recycled
26.5
1.1
3.0
0.7
13.7
Not
Recycled
51.3
1.9
10.7
19.3
78.7
Total
Total
Use the information shown in the table to find each probability.
a) P(Recycled)
b) P(Paper)
c) P(Recycled and Paper)
SOLUTION
a)
b)
c)
P(Recycled) = 45/206.9 = 0.22
P(Paper) = 77.8/206.9 = 0.38
P(Recycled and Paper) = 26.5/206.9 = 0.13
CONDITIONAL PROBABILITY RULE
Try:
a) P(Paper | Recycled)
b) P(Recycled | Aluminum)
c) P(Aluminum | Recycled)
CONDITIONAL PROBABILITY WITH TREE
DIAGRAMS

Only 5% of male high school basketball, baseball,
and football players go on to play at the college
level. Of these, only 1.7% enter major league
professional sports. Sometimes gifted athletes
are able to skip college and play professionally
right out of high school, but this only occurs
about 0.01% of the time.
P(high school athlete goes pro)
 P(pro athlete played at the college level)
 P(pro athlete didn’t play at the college level)
