Discrete Probability - geometric and uniformA fair 20-sided die is rolled repeatedly, until a gambler decides to stop. The gambler receives the amount shown on the die when the gambler stops. The gambler decides in advance to roll the die until a value of m or greater is obtained, and then stop (where m is a fixed integer with 1 &lt; m &lt; 20).(a) What is the expected number of rolls (simplify)?

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Discrete Probability - geometric and uniformA fair 20-sided die is rolled repeatedly, until a gambler decides to stop. The gambler receives the amount shown on the die when the gambler stops. The gambler decides in advance to roll the die until a value of m or greater is obtained, and then stop (where m is a fixed integer with   1  &amp;lt;  m  &amp;lt;  20).(a) What is the expected number of rolls (simplify)?

Micah Hobbs · Accepted Answer

Step 1Let X denote the number of rolls and let A denote the event that at first throw a value ≥m is obtained.Step 2Then:      E    X  =      E    [  X  ∣  A  ]  P  (  A  )  +      E    [  X  ∣      A          ∁        ]  P  (      A          ∁        ).Work this out and a linear equation arises allowing you to find EX.

memLosycecyjz · Accepted Answer

Step 1Let N denote the number of rolls until the first m or a greater number appears. N may take values greater or equal than 1.For the sake of simplicity, let   m  =  18.The probability that   N  =  1 equals the probability that the first roll gives either 18 or 19 or 20, That is   P  (  N  =  1  )  =      3    20    .Step 2The probability that   N  =  2 equals the probability that the first roll is neither 18 nor 19 nor 20 and the third one is. So   P  (  N  =  2  )  =      (    1    −          3      20        )        3    20    .Perhaps, it is clear now that   P  (  N  =  k  )  =            (      1      −              3        20            )              k      −      1            3    20    .There remain a few easy questions:What kind of distribution is this?How to calculate the expectation?Finally, how to generalize the result from 18 to an arbitrary number between 1 and 20?

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