Probability about a geometric distributionBill, Mary and Tom have coins with respective probabilities p 1 , p 2 , p 3 of turning up heads. They toss their coins independently at the same times.What is the probability that neither Bill nor Tom get a head before Mary?

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Probability about a geometric distributionBill, Mary and Tom have coins with respective probabilities       p    1    ,      p    2    ,      p    3   of turning up heads. They toss their coins independently at the same times.What is the probability that neither Bill nor Tom get a head before Mary?

Jaylynn Huffman · Accepted Answer

Step 1Each group of simultaneous tosses has three possible outcomes. If Mary obtains &quot;head&quot;, the game is over with positive outcome. If Bill and/or Tom obtains &quot;head&quot; and Mary doesn&#039;t, the game is over with negative outcome. If no-one obtains &quot;head&quot;, the game continues with the same probabilities as before. Thus, the probability of a positive outcome is the sum of three contributions:                    p                            =                              p          2                ⋅        1        +        (        1        −        (        1        −                  p          1                )        (        1        −                  p          3                )        )        (        1        −                  p          2                )        ⋅        0        +        (        1        −                  p          1                )        (        1        −                  p          2                )        (        1        −                  p          3                )        ⋅        p                                          =                              p          2                ⋅        1        +        (        1        −                  p          1                )        (        1        −                  p          2                )        (        1        −                  p          3                )        ⋅        p                .            Step 2Solving for p yields   p  =            p      2              1      −      (      1      −              p        1            )      (      1      −              p        2            )      (      1      −              p        3            )          .

Bill, Mary and Tom have coins with respective probabilities p_1, p_2, p_3 of turning up heads. They toss their coins independently at the same times. What is the probability that neither Bill nor Tom get a head before Mary?

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