Conditional Probability and Independence
     Learning Targets
   After this lesson, you should be able to:
         Find and interpret conditional probabilities using two-way 
           tables.
         Use the conditional probability formula to calculate 
           probabilities.
         Determine whether two events are independent.
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     Conditional Probability and Independence
     The probability of an event can change if we know that some 
     other event has occurred.
     Conditional Probability
     Conditional Probability
     The probability that one event happens given that another 
     The probability that one event happens given that another 
     event is known to have happened is called a conditional 
     event is known to have happened is called a conditional 
     probability. The conditional probability that event B 
     probability. The conditional probability that event B 
     happens given that event A has happened is denoted by P(B 
     happens given that event A has happened is denoted by P(B 
     | A).
     | A).
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     Happy, Healthy, Rich, or Famous?
     Conditional probabilities and two-way 
     PROBLEM: One question on the Census at School survey (
     tables
     http://www.amstat.org/censusatschool/index.cfm) asks students if 
     they would prefer to be happy, healthy, rich or famous. Students may 
     only choose one of these responses. The two-way table below 
     summarizes the responses of 218 high school students from the 
                                           GENDER        
     United States by gender.
                                        Female  Male   Total
                                Happy    90      46    136
                                Healthy  20      13    33
                        STATUS
                                 Rich    10      31    41
                                Famous   0       8      8
                                 Total   120     98    218
     Suppose we randomly select one of these 218 students. Define 
     events F: female, H: happy, and R: rich.
Statistics and Probability with Applications, 3rd Edition                  44
         Happy, Healthy, Rich, or Famous?
         Conditional probabilities and two-way 
         tables                                                        GENDER                
                                                                  Female       Male       Total
                                                     Happy          90          46         136
                                                     Healthy        20          13         33
                                       STATUS
                                                      Rich          10          31         41
                                                     Famous         0           8           8
                                                      Total        120          98         218
         (a) Find P(H | F).  Interpret this value in context.
         P(H | F) 
         = P(happy | female) 
         = 90/120 
         = 0.75  
         Given that the randomly chosen person is female, there is 
         about a 75% chance that she prefers to be happy. 
         (b) Given that the chosen person did not choose rich, what’s the 
         probability that this person is female?  Write your answer as a 
         probability statement using correct symbols for the events.
                                                             90200 110
         P(female | not rich)  = P(F | RC) =                                          0.814
                                                             136338 177
Statistics and Probability with Applications, 3rd Edition                                                                    55
     Conditional Probability and Independence
     By exploring probabilities through a two-way table, we can 
     determine a general formula for computing conditional 
     probabilities.
     Calculating Conditional Probabilities
     Calculating Conditional Probabilities
     To find the conditional probability P(A | B), use the formula
     To find the conditional probability P(A | B), use the formula
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