probability of becoming a professional athlete by spencerjohnson

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									 Probability
Fundamentals
       Stat 201
Prof. Yanni Papadakis
                  Key Terms
§   Probability Axioms
§   Rules for Addition
§   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

                     Addition Rule
§ 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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