A, B, and C Roll DiceThree players A, B, and C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7 ( P [ 7 ] = 1 / 6 ) on a given roll of the dice. If A rolls 1st, then B, then C, then back to A (etc. etc. etc.). What is the P[A wins], P[B wins], P[C wins]?

Question

A, B, and C Roll DiceThree players A, B, and C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7   (  P  [  7  ]  =  1      /    6  ) on a given roll of the dice. If A rolls 1st, then B, then C, then back to A (etc. etc. etc.). What is the P[A wins], P[B wins], P[C wins]?

i4epdp · Accepted Answer

Step 1Round 1 : A wins with probability 1/6Round 2 : B wins with probability   1      /    6  ×  (  5      /    6  )Round 3 : C wins with probability   1      /    6  ×  (  5      /    6      )    2  Round 4 : A wins with probability   1      /    6  ×  (  5      /    6      )    3  Step 2There is a cube because A has to wait three times for his turn to come again.

chivistaelmore · Accepted Answer

Step 1A wins if they succeed immediately, or if there are first three(!) failures by A, B, C and then a success by A, etc.Alternative approach: If A happens to fail on first roll (with probability       5    6  ) , we essentially start a game with players (B,C,A)  instead of (A,B,C). Therefore   P  [  B  ]  =      5    6    P  [  A  ],   P  [  C  ]  =      5    6    P  [  B  ], and   P  [  A  ]  =      1    6    +      5    6    P  [  C  ]. You can solve these three equations in three unknowns

Three players A, B, & C take turns rolling a pair of dice. The winner is the first player who obtains the sum of 7 (P[7] = 1/6) on a given roll of the dice. If A rolls 1st, then B, then C, then back to A (etc. etc. etc.). What is the P[A wins], P[B wins], P[C wins]?

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