A game is played by rolling a six sided die which has four red faces and two blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or the die has been thrown 4 timesAdnan and Beryl each have one turn. Find the probability that Adnan throws the die more turns than BerylI tried : Adnan throws two times and Beryl throws once = 2 3 x 1 3 Adnan throws three times and Beryl throws once = 4 9 x 1 2 Adnan throws three times and Beryl throws twice = 4 9 x 2 3 Adnan throws four times and Beryl throws once = 8 27 x 1 2 Adnan throws four times and Beryl throws twice = 8 27 x 2 3 Adnan throws four times and Beryl throws three times = 8 27 x 4 9 The answer says 0.365

Question

A game is played by rolling a six sided die which has four red faces and two blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or the die has been thrown 4 timesAdnan and Beryl each have one turn. Find the probability that Adnan throws the die more turns than BerylI tried : Adnan throws two times and Beryl throws once =       2    3   x       1    3  Adnan throws three times and Beryl throws once =       4    9   x       1    2  Adnan throws three times and Beryl throws twice =       4    9   x       2    3  Adnan throws four times and Beryl throws once =       8    27   x       1    2  Adnan throws four times and Beryl throws twice =       8    27   x       2    3  Adnan throws four times and Beryl throws three times =       8    27   x       4    9  The answer says 0.365

Caiden Barrett · Accepted Answer

When Beryl throws the die once, Adnan can throw it 2,3 or 4 times. The required probability is                              1          3                ⏟                    Beryl=1            (    1    −                                        1            3                    ⏟                            Adnan=1              )  Similarly, when Beryl throws the die 2 times, Adnan may throw 3 or 4 times, giving the required probability                                          2            3                                1            3                          ⏟                    Beryl=2            (    1    −          (                                                  1              3                        ⏟                                    Adnan=1                    +                                                                  2                3                                            1                3                                      ⏟                                    Adnan=2                    )        )  and when Beryl throws it 3 times, Adnan throws the die 4 times. The last throw could result in a red face or a blue face. So the probability of this case is                                          2            3                                2            3                                1            3                          ⏟                    Beryl=3            (                                                      2              3                                      2              3                                      2              3                                      [                              2                3                            +                              1                3                            ]                                ⏟                            Adnan=4              )  The sum of these terms yields the required answer.

Ellen Chang · Accepted Answer

Well, A throws it more often than B in the following cases:B throws 1, A throws 2,3,4B throws 2, A throws 3,4B throws 3, A throws 4And now you go and check for the all-together probability of these events. For example:B throws 1, A throws 2,3,4:      P    (  B  =  1  )  =      2    6   and       P    (  A  ∈  {  2  ,  3  ,  4  }  )  =      P    (  A  ≠  1  )  =      4    6  . And finally       P    (  B  =  1  ∩  A  ≠  1  )  =      P    (  B  =  1  )  ⋅      P    (  A  ≠  1  )  =      2    9  . Note that I used that the events are independent.

A game is played by rolling a six sided die which has four red faces and two blue faces. One turn co

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