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Statistics



                      Notes         Alternative Method :

                                    The given problem can be summarised into the following nine-square table:
                                                                          B B Total
                                                                      A A
                                                                       2   2
                                                                A A    6   6 12
                                                                  1
                                                                A  1  2    6  8
                                                              Total    8  12 20

                                    The required probabilities can be directly written from the above table.


                                           Example 29: Two unbiased dice are tossed. Let w denote the number on the first die and
                                    r denote the number on the second die. Let A be the event that w + r  4 and B be the event that
                                    w + r  3. Are A and B independent?
                                    Solution.
                                    The sample space of this experiment consists of 36 elements, i.e., n(S) = 36. Also, A = {(1, 1), (1, 2),
                                    (1, 3), (2, 1), (2, 2), (3, 1)} and B = {(1, 1), (1, 2), (2, 1)}.
                                    From the above, we can write

                                                      6   1       3   1
                                                P A       , P B    
                                                              ( )
                                                 ( ) 
                                                      36  6       36  12
                                                                          3   1
                                     Also  (A   B ) {(1,1),(1,2),(2,1)}       P  (A  B )  
                                                                          36  12
                                    Since  P A B b  g b g
                                                 
                                                   P A P B b g , A and B are not independent.
                                           Example 30: It is known that 40% of the students in a certain college are girls and 50% of
                                    the students are above the median height. If 2/3 of the boys are above median height, what is the
                                    probability that a randomly selected student who is below the median height is a girl?
                                    Solution.

                                    Let A be the event that a randomly selected student is a girl and B be the event that he/she is
                                    above median height. The given information can be summarised into the following table :

                                                                          B   B Total
                                                                   A    10 30   40
                                                                   A    40 20   60
                                                                  Total  50 50 100

                                                                       )
                                    From the above table, we can write  ( /P A B   30    0.6 .
                                                                          50


                                           Example 31: A problem in statistics is given to three students A, B and C, whose chances
                                                               1 1    1
                                    of solving it independently are   ,  and    respectively. Find the probability that
                                                               2 3    4
                                    (a)  the problem is solved.
                                    (b)  at least two of them are able to solve the problem.



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