When A and B flip coins, the one coming closest to a given line wins 1 penny from the other. If A starts with 3 and B with 7 pennies, what is the probability that A winds up with all of the money if both players are equally skilled? What if A were a better player who won 60% percent of the time?

Question

Franklin Frey · Accepted Answer

Step 1For   1  ≤  n  ≤  9, let       p    n   be the probability that A wins if A has n pennies before starting the next round.Then we have the following system of 5 equations in 5 unknowns ...      {                                        p            1                    =                      1            2                                p            2                                                            p            2                    =                      1            2                                p            3                    +                      1            2                                p            1                                                            p            3                    =                      1            2                                p            4                    +                      1            2                                p            2                                                            p            4                    =                      1            2                                p            5                    +                      1            2                                p            3                                                            p            5                    =                      1            2                                                  [by symmetry]                        Solving the system yields       p    3    =      3    10  .Step 2If instead of equal skills, we assume that A has probability       3    5   of winning each round, we get the following system of 9 equations in 9 unknowns...      {                                        p            1                    =                      3            5                                p            2                                                                                                              p            2                    =                      3            5                                p            3                    +                      2            5                                p            1                                                            p            3                    =                      3            5                                p            4                    +                      2            5                                p            2                                                            p            4                    =                      3            5                                p            5                    +                      2            5                                p            3                                                            p            5                    =                      3            5                                p            6                    +                      2            5                                p            4                                                            p            6                    =                      3            5                                p            7                    +                      2            5                                p            5                                                            p            7                    =                      3            5                                p            8                    +                      2            5                                p            6                                                            p            8                    =                      3            5                                p            9                    +                      2            5                                p            7                                                            p            9                    =                      3            5                    +                      2            5                                p            8                                  Solving the system yields       p    3    =                              41553          58025                      ≈  0.7161223611

When A and B flip coins, the one coming closest to a given line wins 1 penny from the other. If A starts with 3 and B with 7 pennies, what is the probability that A winds up with all of the money if both players are equally skilled? What if A were a better player who won 60% percent of the time?

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