Transcript Statistics

Statistics
General Probability Rules
Union

The union of any collection of events
is the event that at least one of the
collection occurs
Addition Rule for Disjoint Events
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If events A, B, and C are disjoint in
the sense that no two have any
outcomes in common, then
P(one or more of A, B, C) = P(A) + P(B) + P(C)
Not Disjoint
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If events A and B are not disjoint,
they can occur simultaneously.
P(A or B) = P(A) + P(B) – P(A and B)
P( A  B)  P( A)  P( B)  P( A  B)
Deborah and Marshall problem
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P 362, example 6.17
Deborah = 0.7
Matthew = 0.5
Both = 0.3
Question – What is the probability that at least one of them is
promoted?
What is the probability of at least
one of them is promoted?
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P(at least one) = 0.7 + 0.5 – 0.3
P(at least one) = 0.9
Table of probabilities
Promoted
Deb Promoted 0.3
Not
promoted
total
0.7
Not
promoted
Total
0.5
1.0
Questions
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P(D and M) =
P(D and not M) =
P(Not D and M) =
P(Not D and not M) =
Answers
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P(D and M) = 0.3
P(D and not M) = 0.4
P(Not D and M) = 0.2
P(Not D and not M) = 0.1
Problems to do
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46, 53
Conditional Probability
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
the probability of an event happening
knowing that another event has
happened.
Written as P(AlB) the probability of
B happening knowing that A has
happened.
Married
18-29
30-64
65+
total
7,842
43,808
8,270
59,920
7,184
751
21,865
2,523
8,385
10,944
Never
13,930
Married
Widowed 36
Divorced
704
9,174
1,263
11,141
Total
22,512
62,689
18,669
103,870
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A = the woman chosen is young,
ages 18 to 29
B = the woman chosen is married
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P(A) = 22,512/103,870 = 0.217
P(A and B) = 7,842/103,870 = 0.075
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Probability she is married given that
she is young.
P(B l A) = 7,842/22,512 = 0.348
General multiplication rule for two
events
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P(A and B) = P(A)P(B l A)
Definition of Conditional Probability
P( AandB)
P( B | A) 
P( A)
Problems
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56, 58
Extended Multiplication rules
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Intersection: the intersection of any
collection of events is the event that
all of the events occur.
Example

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The intersection of three events A, B,
and C has the probability
P(A and B and C) =
= P(A)P(B|A)P(C|A and B)
Future of High School Athletes
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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.
About 40% of the athletes who
compete in college and then reach
the pros have a career of more than
3 years.
Events
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A = {competes in college}
B = {competes professionally}
C = {pro career longer than 3 years}
P(A) = 0.05
P(B|A) = 0.017
P(C|A and B) = 0.4
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P(A and B and C) =
= P(A)P(B|A)P(C|A and B)
= 0.05 x 0.017 x 0.40
= 0.00034
Only 3 out of every 10,000 high
school athletes can expect to
compete in college and have a career
greater than 3 years
Tree Diagrams
The probability P(B) is the sum of the probabilities of the two
branches ending at B.
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Probability of
reaching B given
college is
0.05x0.017=0.00085
Probability of reaching
B not going to college
is
0.95x0.0001=0.000095
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Probability of P(B) =
0.00085+0.000095
= 0.000945
Or about 9 students out of 10,000
will play professional sports.
Problems
64, 67, 70, 77, 79, 80, 83, 87