An experiment is conducted until it results in success: the first step has probability 1 2 to be successful, the second step (only conducted if the first step had no success) has probability 1 3 to be successful, the third step (only conducted if the first two steps had no success) has probability 1 4 to be successful : if none of the steps were successful, we repeat the experiment until success is achieved. Assuming the first step has cost 2, and the second step as well as the third step have cost 1, what is the overall cost until success ?

Question

An experiment is conducted until it results in success: the first step has probability       1    2   to be successful, the second step (only conducted if the first step had no success) has probability       1    3   to be successful, the third step (only conducted if the first two steps had no success) has probability       1    4   to be successful : if none of the steps were successful, we repeat the experiment until success is achieved. Assuming the first step has cost 2, and the second step as well as the third step have cost 1, what is the overall cost until success ?

Irene Simon · Accepted Answer

Step 1If   X  =  4, the cost will be   2  +  1  +  1  +  2  =  6. You must take into account that X can be any natural number.Step 2To fix this, let Y be the cost, and take your equation   E  (  Y  )  =  2  ⋅  1  ⋅  P  (  X  =  1  )  +  1  ⋅  2  ⋅  P  (  X  =  2  )  +  1  ⋅  3     P  (  X  =  3  ) and modify it to   E  (  Y  )  =  2  ⋅  P  (  X  =  1  )  +  3  ⋅  P  (  X  =  2  )  +  4  ⋅     P  (  X  =  3  )  +  q  (  4  +  E  (  Y  )  ) and modify it to   E  (  Y  )  =  2  ⋅  P  (  X  =  1  )  +  3  ⋅  P  (  X  =  2  )  +  4  ⋅     P  (  X  =  3  )  +  q  (  4  +  E  (  Y  )  ) where   q  =      1    2    ⋅      2    3    ⋅      3    4   is the probability of failure on all three steps, and the   (  4  +  E  (  Y  )  ) comes from the fact that if we fail the three steps, we are back where we started, plus we have already spent a cost of 4. (I have also modified the coefficients as per lulu&#039;s comment.)

darcybabe98ub · Accepted Answer

Step 1Let       μ          0       denote the expectation of the cost of the steps yet to be done if no steps have been made.Let       μ          1       denote the expectation of the cost of the steps yet to be done if step 1 has been made without success.Let       μ          2       denote the expectation of the cost of the steps yet to be done if step 1 and step 2 have been made without success.Step 2Then we have the following equalities.      μ          0        =      1    2    2  +      1    2        [    2    +          μ              1              ]    =  2  +      1    2        μ          1            μ          1        =      1    3    1  +      2    3        [    1    +          μ              2              ]    =  1  +      2    3        μ          2            μ          2        =      1    4    1  +      3    4        [    1    +          μ              0              ]    =  1  +      3    4        μ          0      Then:       μ    0    =  2  +      1    2        [    1    +          2      3              [      1      +              3        4                    μ        0            ]        ]  The solution of this equality is:       μ    0    =      34    9

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