Geometric distribution expected number of rollsA pair of dice are repeatedly rolled until the two sum to ≥ 10. The expected number of times the pair is rolled is?I understand how to apply the geometric to get the probability of an event, but I do not understand how to use it to get the expected number of rolls (in this case) to achieve a certain result.

Question

Geometric distribution expected number of rollsA pair of dice are repeatedly rolled until the two sum to   ≥  10. The expected number of times the pair is rolled is?I understand how to apply the geometric to get the probability of an event, but I do not understand how to use it to get the expected number of rolls (in this case) to achieve a certain result.

Malcolm Mcbride · Accepted Answer

Step 1First, the probability that two dice sum up to   ≥  10 is 16 since there are six possible events, {46,55,56,64,65,66}. Thus, if X counts the number of rolls, then   Pr  (  X  =  k  )  =      1    6  . You can now simply use the fact that       E    (  X  )  =      1    p   and get 6. Alternatively, here is the calculation:      E    (  X  )  =      ∑          k      =      1        ∞    k  ⋅  Pr  (  X  =  k  )  =      ∑          k      =      1        ∞    k  (      5    6        )          k      −      1            1    6    =      1    6        ∑          k      =      0        ∞    k  (      5    6        )          k      −      1      Step 2Let   f  (  x  )  =      x          1      −      x      . Then for   x  ∈  (  0  ,  1  ), we have   f  (  x  )  =      ∑          k      =      0        ∞        x    k  . Thus,       ∑          k      =      0        ∞    k      x          k      −      1        =      f    ′    (  x  )  =      1          (      1      −      x              )        2            . Therefore,       E    (  X  )  =      1    6        f    ′    (      5    6    )  =      1    6        1          (              1        6                    )        2              =  6.

A pair of dice are repeatedly rolled until the two sum to ge 10. The expected number of times the pair is rolled is?

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