# probability of becoming a professional athlete by spencerjohnson

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Probability
Fundamentals
Stat 201
Key Terms
§   Probability Axioms
§   Rules for Multiplication
§   Venn Diagram
§   Tree Diagram
§   Bayes’ Rule
Terminology
Example: Throw of 2 6-face dice
§ Outcome (? ): e.g., (5,2)

§ Sample Space (O): all 36 outcomes
{(1,1),(1,2),(1,3),…,(6,5),(6,6)}

§ Event (A): Any subset of O
e.g. sum of 5 {(1,4),(2,3),(3,2),(4,1)}
or at least one die is 5
{(1,5),(2,5),(3,5),(4,5),(5,5),(6,5),
(5,1),(5,2),(5,3),(5,4),(5,6)}
Venn Diagram       Set Notation       Logical Meaning

ω∈A
A

?                                ? realizes A
belongs

B
A
A∩ B = ∅                A and B
Intersection     are Incompatible
Equals Null Set

B
A
A⊂ B
A implies B
Subset
Venn Diagram         Set Notation             Logical Meaning
B           A∪ B − A∩ B
One And Only One
A
A or B − A and B            Event of A and B
is Realized
Union minus Intersection

B
A∩ B
A                                   Both A and B
A or B                 are realized
Intersection

O
B           A   A+ B +C = Ω
A,B,C are
A∩ B = ∅
exhaustive and
C
A∩C = ∅             mutually exclusive
B ∩C = ∅
Venn Diagram    Set Notation           Logical Meaning
O
A        A’   A' = Ω − A
All but A
Complement              is realized

B
A∩ B A| B =   A ∩ B but Ω = B     A given B
is realized
Given
Work with partner
§ Common sources of caffeine in diet are coffee, tea, and
cola drinks. Suppose that
§ 55% of adults drink coffee
§ 25% of adults drink tea
§ 45% of adults drink cola
§ Also
§   15% drink both coffee and tea
§   5% drink all three beverages
§   25% drink both coffee and cola
§   5% drink tea only
§ Draw Venn Diagram
§ What percent drinks cola only?
§ What percent drinks none of the above beverages?
Probability Rules
§ P(A) = 0   P(A) = 1
§ P(O) = 1   P(Æ) = 0

§ P(A + B) = P(A) + P(B) - P(A and B)
P(A + B) = P(A) + P(B)      For Mutually   Exclusive Events
P(A’) = 1 - P(A)

§ P( A|B ) = P(A and B) / P(B)        ,or

Multiplication Rule
§ P(A and B) = P(A|B) P(B)
Combining Outcomes
§ A: First die shows 4
One die sample space is {1,2,3,4,5,6}
P(A)=1/6
§ B: Second die shows 5
Other die sample space is {1,2,3,4,5,6}
§ Throw two dice together
New space {(1,1),(1,2),…,(6,6)}
§ Extend first die space, then Event is
{(4,1),(4,2),…,(4,6)}
P(A) = 6/36 = 1/6
Event Independence
§ P(A and B) = P(A) P(B)   INDEPENDENCE TEST 1
but
§ P(A and B) = P(A|B) P(B)
thus
§ P(A|B) = P(A)            INDEPENDENCE TEST 2

§ P(A and B) = P((4,5)) = 1/36

§ P(A) = 6/36   [first die shows 4]
§ P(B) = 6/36   [second die shows 5]
Not Independent Events Example
§ [dice sum to 5]        P(A) = 4/36
§ [second die shows 5]   P(B) = 6/36

§ P(A|B) = 0
but
§ P(A) P(B) = 24/362 = 2 / 108
Probability of Becoming a
Professional Athlete
§ The probability a male high school athlete will go
to college is 5%. If they do not go to college, then
they have a chance of 0.01% to become
professional athletes. Of those who do go to
college, 98.3% do not continue their career as
professional athletes.
§ What are the odds for a male high school athlete
to make it to professional sports?
§ Draw the relevant tree diagram
§ How do you explain the tree diagram to a high
school kid contemplating a career in professional
sports?
Problem 5.15
Gas Well Completions 1986         D        D’
Dry    Not Dry

N         North America         14,131   31,575    45,706
N’        South America           404    2,563     2,967
14,535   34,138    48,673

Draw Venn Diagram
Identify Event ( N and D )
What is its probability?
Probability Tree: Problem 5.15

• Location first
D   14,131/45,706
N                                    14,131/48,673

45,706/48,673      D’ 31,575/45,706
31,575/48,673

D     404/2,967       404/48,673
N’

2,967/48,673        D’ 2,563/2,967      2,563/48,673

Are Events N and D Independent?
What should the number of dry wells in So. America be,
for N and D to be independent?
Bayes’ Rule (form 1)

P( A and B)
P( A | B) =
P( B)
P( B | A) P( A)
P( A | B) =
P( B | A) P( A) + P( B | A' ) P( A' )
Bayes’ Rule (form 2)

P( A and B)
P( A | B) =
P( B)
P( B | Ai ) P( Ai )
P( Ai | B) =
∑ P( B | Ai ) P( Ai )
i

where A i disjoint and exhaustive subsets of Ω
HS Athlete          College              Professional

B|A    0.05x0.017= 0.00085
0.017

A

0.05
0.983           B’|A   0.05x0.983= 0.04915

B|A’    0.95x0.00001=0.0000095
0.0001
0.95
A’

0.9999
B’|A’   0.95x0.9999=0.949991
P(A|B)=?
Work on this in groups
§ To detect naileria (an imaginary nail disease),
doctors apply a test, which, if a patient really
suffers from the disease, gives a positive result
with probability 99%. However, it may happen
that a healthy subject obtains a positive result.
This probability of false positives is 2%. It is
known that the prevalence (frequency of
occurrence in the population) of naileria is 0.1%.
What is the probability that a patient who gets a
positive on the test really suffers from naileria?
Assume the test is administered to the population
at large and not only to patients with naileria
symptoms.
Albinism Example
§  People with Albinism have little pigment in their skin, hair,
and eyes. The gene that governs albinism has two forms
(called alleles), denoted as a and A. Each person has a
pair of alleles inherited from parents with probability 0.5.
Albinism gene is recessive, one has disease only if both
alleles are aa.
1. Beth’s parents are not albino but she has an albino
brother. This implies both her parents are type aA. Why?
2. Which of the type aa, Aa, AA could a child of Beth’s
parents have? What is the probability of each type?
3. Beth is not an albino. What are the conditional
probabilities for Beth’s possible genetic types, given this
fact?
Albinism continued
§  Beth knows the probabilities for her genetic types
from previous analysis. She marries Bob, who is
albino (type aa).
1. What is the conditional probability that Beth and
Bob’s child is non-albino, if Beth is type Aa? What
is the answer to the latter if Beth is type AA?
2. Beth and Bob’s first child is non-albino. What is
the conditional probability that Beth is a carrier,
type Aa?
§ Solve using tree diagram
Work on this in groups
§ Factories A and B manufacture watches. Factory A
produces defectives on average at rate 1/100.
Factory B produces defectives at rate 1/200. A
retailer receives a case of watches from one the
above factories, without knowing from which one
(assume equal probabilities). She checks the first
watch and it works.
§ What is the probability that the second watch in
the case works?
§ Is this independent from the first watch working?
Card Magic
§ A magician shuffles three cards: one with
two red faces, one with two white faces,
and one with one white and one red face.
She randomly selects a card, places it on
the table and shows its open face being red.
§ What is the probability that the covered face
is also red?
§ No, the answer is not 50%

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