Transcript SIA_Ch_5.5_Notes

Section 5.5 - Independent Events
Objectives:
1.
Understand the definition of independent events.
2.
Know how to use the Multiplication Rule for
Independent Events.
3.
Understand that real data rarely meet the mathematical
definition of independence.
4.
Develop insight into when it is reasonable to assume
independence as part of the model.
Section 5.5 - Independent Events
Example: Tap Water and Bottled Water
In Jack and Jill’s tests with tap and bottled water, they found that how well a
person did depended on whether that person regularly uses bottled water.
• Of the people who regularly drink bottled water, 24 / 30 = 80%
correctly identified tap water.
• Of the people who drink tap water, only 36 / 70 = 51.4% correctly
identified tap water.
Display 5.48
Drinks
Bottled
Water?
Identified Tap Water
Yes
No
Total
Yes
24
6
30
No
36
34
70
Total
60
40
100
Section 5.5 - Independent Events
Example: Tap Water and Bottled Water
On the other hand, men and women did equally well in identifying tap water
• Men: 21 / 35 = 60% correctly identified tap water.
• Women: 39 / 65 = 60% correctly identified tap water.
• Overall: 60 /100 = 60% correctly identified tap water.
Display 5.49
Gender
Identified Tap Water
Yes
No
Total
Male
21
14
35
Female
39
26
65
Total
60
40
100
Section 5.5 - Independent Events
Example:
Tap Water and Bottled
Water
The events “drinks bottled
water” and “correctly
identifies tap water” are
dependent events.
The events “is a male” and
“correctly identifies tap
water” are independent
events. Being male or
female doesn’t change
the probability that a
person correctly
identifies tap water.
Identified Tap Water
Display 5.48
Drinks
Bottled
Water?
Yes
No
Total
Yes
80%
20%
100%
No
51.4%
48.6%
100%
Total
60%
40%
100%
Display 5.49
Gender
Identified Tap Water
Yes
No
Total
Male
60%
40%
100%
Female
60%
40%
100%
Total
60%
40%
100%
Section 5.5 - Independent Events
Definition of Independent Events
Suppose P(A)  0 and P(B)  0.
Events A and B are independent if and only if
P(A)  P(A | B) or, equivalently, P(B)  P(B | A)
P(A)  0 and P(B)  0 
A and B are independent  P(A)  P(A | B).
Section 5.5 - Independent Events
Definition: If and Only If
S is true if and only if T is true.
If S is true, then T is true.
If T is true, then S is true.
S <=> T
S => T
T => S
Example:
A man may vote if and only if he is has registered to vote. V <=> R
If a man may vote, then he has registered to vote. V => R
If a man has registered to vote, then he may vote. R => V
If a man may vote, then he is a U.S. citizen. V => C
However, if a man is a U.S. citizen, it is not true that he may vote. (He
must be registered.)
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Two events A and B are independent if and only if
P(A and B)  P(A  B)  P(A)  P(B)
More generally, events A1, A2, …, An are independent if and
only if
P(A1  A2  ...  An )  P(A1 )  P(A2 )  ... P(An )
n
 n

P  An    P Ai 
 i1  i1
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
This is both a rule for computing probabilities:
A and B are independent  P(A  B)  P(A)  P(B)
and a definition of independence
P(A  B)  P(A) P(B)  A and B are independent
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Four Flips
If you flip a fair coin four times, what is the probability of four
heads?
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Four Flips
If you flip a fair coin four times, what is the probability of four
heads?
HHHH HHHT HHTH HTHH THHH
HHTT TTHH HTHT THTH HTTH THHT
TTTT TTTH TTHT THTT HTTT
P(HHHH) = 1/16
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Four Flips
If you flip a fair coin four times, what is the probability of four
heads?
The outcomes of the first flips don’t change the
probabilities on the remaining flips, so the flips are
independent.
P(HHHH )  P(H )  P(H )  P(H )  P(H )
4
1
 1  1  1  1  1
       
 2  2  2  2  2
16
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Health Insurance
About 30% of adults ages 19 to 29 don’t have health
insurance. What is the chance that if you choose two
adults from this age group at random, the first has
insurance and the second doesn’t?
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Health Insurance
About 30% of adults ages 19 to 29 don’t have health
insurance. What is the chance that if you choose two
adults from this age group at random, the first has
insurance and the second doesn’t?
A small sample is selected from a large population, so it is
appropriate to model the outcomes as independent events.
P(insurance  yes)  0.70
P(insurance  no)  0.30
P((yes, no))  (0.70)(0.30)  0.21
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Health Insurance
About 30% of adults ages 19 to 29 don’t have health
insurance. What is the chance that if you choose two
adults from this age group at random, the first has
insurance and the second doesn’t?
Display 5.50
Has Insurance?
1st Young
Adult
2nd Young Adult
No
Yes
Total
No
0.09
0.21
0.30
Yes
0.21
0.49
0.70
Total
0.30
0.70
1.00
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Computing the Probability of “At Least One”
Suppose you take a random sample of ten adults from this
age group. What is the probability that at least one of them
doesn’t have insurance?
Section 5.5 - Independent Events
The Multiplication Rule for Independent Events
Example: Computing the Probability of “At Least One”
Suppose you take a random sample of ten adults from this
age group. What is the probability that at least one of them
doesn’t have insurance?
The event “at least one doesn’t have insurance” is the
complement of the event “all have insurance”
P(YYYYYYYYYY )  (0.70)10  0.028
P(at least one N )  1  P(all Y )  1  0.028  0.972
Section 5.5 - Independent Events
Independence with Real Data
Example: Independence and Baseball
In a recent season, by July 1 the L.A. Dodgers had won 41
games and lost 37. The breakdown by day / night is shown
below in Display 5.51. If one of these games is chosen at
random, are the events “win” and “day game”
independent?
Won the Game?
Time of
Game
Yes
No
Total
Day
11
10
21
Night
30
27
57
Total
41
37
78
Section 5.5 - Independent Events
Independence with Real Data
Example: Independence and Baseball
Are the events “win” and “day game” independent?
P(win) = 41 / 78 = 0.526
P(win | day game) = 11 / 21 = 0.524
Since the two probabilities are not exactly equal, the
events are not independent.
Section 5.5 - Independent Events
Independence with Real Data
Example: Independence and Baseball
Are the events “win” and “day game” independent?
P(win) = 41 / 78 = 0.526
P(win | day game) = 11 / 21 = 0.524
Since the two probabilities are not exactly equal, the events are not
independent.
Given that the Dodgers played 21 day games, the two percentages couldn’t
be any closer: (12 / 21 = 0.571). It is impossible for the two percentages
to be the same!
The correct conclusion is that if these results can be considered a
random sample of games, there isn’t sufficient evidence to say that
the events are not independent.