Download Chapter 6 Review

Chapter 6 Review
Fr Chris Thiel
13 Dec 2004
What is true about
probability?
• The probability of any event must be a
number between 0 and 1 inclusive
• The sum of all the probabilities of all
outcomes in the sample space must be 1
• The probability of an event is the sum of the
outcomes in the sample space which make
up the event
Independent
Iff P(B | A)  P(B)
Previous outcomes do not change probability
Multiplication Rule: P(A and B)=P(A)P(B)
Disjoint
One outcome precludes the other since there is
No overlap…

Complement
A  A  notA  A
c
The event A does not occur
P(A ) 1 P(A)
c
Addition Rules
P(A or B)=P(A)+P(B)-P(A and B)
Multiplication Rules
P(A and B)=P(A)P(B) if A and B are independent
Conditional Rules
P(A  B)  P(A)P(B | A)
P(A  B)
P(B | A) 
P(A)
P(65+)=18%
P(Widowed)=10%
a. If among 65+, 44% widowed, What percent of the
population are widows over 65?
P(65  W )  P(65)P(W | 65)
 (.18)(.44)  .0792

b. If 8% are widows over 65, What is the chance
of being a widow given that they’re over 65?
P(W  65)
P(W 
| 65) 
P(65)

.08
 .44
.18
See Table 6.1 p. 366
Use Venn Diagrams &
Trees
Venn Diagrams can help see if events are
Independent, complementary or disjoint
Use Tree Diagrams to Organize addition and
Multiplication rules to combinations of events
If event A and B are
disjoint, then
• P(A and B)= 0
• P(A or B) =1
• P(B)=1-P(A)
Independent events…
you flip a coin and it’s heads
4 times in a row…. The odds
are STILL the same
The 6 is 3 times more
likely to occur… what is
the probability of rolling a 1
or a 6?
x  x  x  x  x  3x  1
3
8

 18 
1
2
A fair die is tossed
4 or 5-win $1
6-win $4 If you play twice:
what is the probability that you will win
$8?
$2?
P(A)=.5
P(B)=.6
P(A andB)=.1
•
•
•
•
•
A
B
.4
.1
.5
P(A|B)=?
Are A and B Independent?
Disjoint?
Will either A or B always occur?
Are A and B complementary?
0
Lie Detector
• Reports “Lie” 10% if person is telling
the truth
• Reports “Lie” 95% if the person is
actually lying
• Probability of machine never reporting
a lie if 5 truth tellers use it
(.9)  .59049
5
You enter a lottery, the
odds of getting a prize is
.11
If you try 5 times, what is
the probability that you will
win at least once?
• 1-P(never winning)
1 (.89)
5
8% have a disease. A test detects the disease 96%
And falsely indicates the disease 7%. If you test
positive, what is the chance you have the disease?
Tests +
.96
(.08)(.96)=.0768
.04
Tests -
(.08)(.04)=.0032
Tests +
.07
(.92)(.07)=.0644
.93
(.92)(.93)=.85
Has Disease
.08
.92
No Disease
Tests -


P(D|+)
P(D  )
P(D | ) 
P()
(.0768)

 .65
(.0768)  (.0644)
P(Harvard)=40%
P(Florida)=50%
P(both)=20%
P(none)=?
P(F but not H)=?
H
F
.2
.2
.3
.3
30% of calls result in a airline reservation.
a. P(10 calls w/o a reservation)=?
(1 .3)  .0282
10
b. P(at least 1 out of 10 calls has a reservation)=?
1 P(none) 1 .0282  .9718

85% fire calls are for medical emergencies
Assuming independence…
P(exactly one of two calls is for a medical emergency)=?
P(M)P(F)+P(F)P(M)=(.85)(.15)+(.15)(.85)=.255
Is it really independent?