Transcript Geometry: Statistics 12.2

Geometry: Statistics 12.2
Consider drawing a card from a standard deck of 52
cards. What is the probability that it is a heart?
13 1

52 4
Now consider drawing another card from the deck.
What is the probability that it is a heart?
The answer to this question depends on what you do
with the first card.
If you replace it in the deck…
13 1

52 4
If you do not replace it in
the deck…
12 4

 .235
51 17
Two events are independent events if the
occurrence of the one event does not
affect the occurrence of the other event.
If events are not independent then they are dependent.
Example 1: Are the two events described independent?
You flip a coin and then roll a 6-sided die
Event A: The coin is a head
independent
Event B: The die shows a 3
Event A: Your alarm fails to go off in the morning
Event B: You are late for school
dependent
Your drawer contains the following pairs of socks: 3 black, 4 blue and 2 brown.
You reach in the drawer and randomly choose a pair of socks for you and your
friend.
Event A: The first pair of socks is black
dependent
Event B: The second pair of socks is brown.
Probability of Independent Events: If two
events A and B are independent then,
___________________
P(A and B)  P(A)  P(B)
Note: The converse of this statement is also
true.
Example 2: You flip a coin and then roll a 6-sided die. Find
Since the events are independent
P(head and a 3).
H1 H2 H3 H4 H5 H6
T1 T2 T3 T4 T5 T6
1 1 1
 
2 6 12
Example 3: You spin the spinner at the right twice. Find the
probability you get a 1 on the first spin and 4 or more on the
second spin
Since the events are independent
1 6
6
 
 .042
12 12 144
Example 4: A group of four students includes one boy and
three girls. A teacher randomly selects one of the students to
be the speaker and a second to be the recorder. Show that the
two events A and B are dependent events.
Event A: a girl is chosen as the speaker
Event B: a girl is chosen as the recorder
P( A and B)  P(A)  P(B)
6
12
9
12
.5  .5625
9
12
BG1
G1B
G2B
G3B
BG2 BG3
G1G2 G1G3
G2G1 G2G3
G3G1 G3G2
Two events A and B are dependent if the
occurrence of one DOES affect the occurrence of
the second.
The probability that event B occurs given that
event A has already occurred is called the
______________________________
conditiona
l probability of B given A and is
written as _______
P(B | A)
Example 5: Find P(girl is chosen as recorder | given a girl
was chosen as speaker)
BG1 BG2 BG3
6 2
9
3
G1B G1G2 G1G3
G2B G2G1 G2G3
G3B G3G1 G3G2
Example 6: Consider rolling two 4-sided dice.
P(sum of 6 | exactly one die shows a 4)
1 -1
2 -1
3 -1
4 -1
1- 2 1- 3 1- 4
2-2 2-3 2-4
3- 2 3-3 3- 4
4-2 4-3 4-4
2 1

6 3
Consider the table at the right which shows the number of species in
the U.S. as endangered or threatened.
Example 7: Find P(species is bird | species is endangered)
80
 .171
468
Example 8: Find P(species is threatened | species is mammal)
16
 .186
86
86
468
Probability of Dependent Events: If two events
A and B are dependent events then,
P_______________
(A and B)  P(A)  P(B | A)
Example 9: Consider drawing two cards from a standard
deck of 52 cards. Find P(both cards are hearts)
13
52
  .059
12
51
Example 10: A bag contains 20 $1-bills, 10 $5-bills, 5
$10 bills and 1 $20-bill. You randomly select two bills
from the bag. Find P(first bill is a $1 and the second bill
is a $10)
20
.079


36
5
35
Example 11: A bag contains 20 $1-bills, 10 $5-bills, 5 $10 bills
and 1 $20-bill. You randomly select two bills from the bag.
Find P(first bill is a $5 and the second bill is a $10)
10 5
  .040
36 35
Example 12: Consider drawing three cards from a standard
deck of 52 cards.
Find P(all 3 cards are hearts)
13 12 11
   .013
52 51 50
We can rewrite our formula for dependent events to
obtain a rule for finding conditional probabilities.
P(A) x P(B | A) = P(A and B)
Divide both sides by P(A)
P( A and B)
P( B | A) 
P(A)
This is particularly useful when you are given percents
as opposed to the actual counts.
Example13: At a particular school, 40% of the students
buy breakfast at school, 72% buy lunch at school and
18% buy both breakfast and lunch at school.
Find P(student buys breakfast | student buys lunch)
P( B and L) .18
P ( B | L) 

 .25
P(L)
.72