Conditional Probability
Section 3.6

Public Opinion Polls may categorize respondents by sex, age, race and level of education.  Comparisons are made and trends observed by using conditional probability.

The conditional probability of event A given event B is:  

        P(A | B) = n(A B)
                                        
                            n(B)

Example 1: 

Asked 500 men and women, “Should the driving age be postponed to 18 years?”

(Hypothetical)

Find:

a)     P( Y )                      

b)     P( Y | W )

c)      P( Y | M )

d)     P( M | Y )

e)  P( Y M)

a)  P(Y) = n(Y)/n(S)    = 295/500 = .59 = 59%

b)     P(Y | W) = n(Y  W ) =  185 = .66 = 66%
                     n (W)            280

c)      P( Y | M) = n ( Y   M) = 110  = .50 = 50 %
                        n(M)            220

d)     P(M | Y) = n ( M   Y) =  110  = .37 = 37%
                          n(Y)           295

e)     Requires a new formula not yet derived called the Product Rule.

Recall:       P( Y |  M) = n( Y   M)        then       n(M) *P(Y | M) = n( Y   M)
                                              n(M)

n(M) *P(Y | M)  =  n( Y   M)                           P(M)*P(Y | M) = P( Y   M)
  n(S)                         n(S)

P( Y M) = 220/500 * 110/220 = 110/500 = .22 = 22%

Example 2: 

Find the probability that 2 cards dealt are both aces.

 Let 

        B = 1^st card Ace, 

let 

        A = 2^nd card Ace.  

Find     P (A   B).

        P( A   B) = P(A | B) * P(B)  =  3/51 * 4/52 

        = 3/663 = .0045 =  .45% (less than ½ %)

TREE DIAGRAM:

                                                ACE                            (4/52)(3/51) = .0045

                                    3/51

                        ACE

            4/52                48/51

            NOT ACE                   (4/52)(48/51) = .0723

Start                                                    ACE                (48/52)(4/51) = .0723

            48/52                          4/51  

                        NOT ACE

                                                47/51

                                                            NOT ACE       (48/52)(47/51) = .8506

a)  What is P(B A )?  Top Branch = .0045

b)     What is P (B’  A)?  Third Branch = .0723

c)      What is P(A)?      = P(B A ) + P(B’ A )     

Add Top branch and third branch = .0045 + .0723 = .0768

REVIEW EXAM #3

3.2

Basic Terms of Probability: Experiment, Sample Space, Event, Probability, Odds, Mutually exclusive.  Use in Genetics.

3.3

Basic rules of Probability:  0  P(E)  1; P(S) = 1; P( ) = 0; (6 RULES)

        P(E) = n(E)/n(S) = success/total

        Odds(E) = n(E): n(E’) = success: failure

3.4

Combinatorics and Probability:  Using Permutations and Combinations to find probability.  5-card hands in poker, Lottery.

3.5

Expected Value: ($)P(Win) + (-$)P(Lose) Casino games, Decision Theory.

3.6

Conditional Probability: Conditions on results.  Probabibility of A given B.

P(A|B) = n (A B) / n(B)

P (A B ) = P(A | B )* P(B)

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