Geometric Probability QMike and John are playing a friendly game of darts where the dart board is a disk with radius of 10in. Whenever a dart falls within 1in of the center, 50 points are scored. If the point of impact is between 1 and 3in from the center, 30 points are scored, if it is at a distance of 3 to 5in 20 points are scored and if it is further that 5in, 10 points are scored. Assume that both players are skilled enough to be able to throw the dart within the boundaries of the board. Mike can place the dart uniformly on the board (i.e., the probability of the dart falling in a given region is proportional to its area). (a) What is the probability that Mike scores 50 points on one throw? (b) What is the probability of him scoring 30 points on one throw? (c) John is right handed and is twice more likely to throw in the right half of the board than in the left half. Across each half, the dart falls uniformly in that region. Answer the previous questions for John’s throw

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Sauppypefpg · Accepted Answer

Step 1The probability for both players are the same!Player 1: P(50 points) =             1      2              10      2        =      1    100  , as you noted.Step 2Player 2: P(50 points) =       2    3                      1        2                    /            2                      10        2                    /            2        +      1    3                      1        2                    /            2                      10        2                    /            2        =      1    100    ..Similarly for each other score.

Antwan Perez · Accepted Answer

Step 1The probability of landing inside a circle of radius r is propotional to the area of that circle.  P  (  x  &amp;lt;  r  )  =  C  π      r    2  To find that constant C, we can use the fact that   P  (  x  &amp;lt;  R  ) should be 1.This implies that   C  π      R    2    =  1 and that   C  =      1          π              R        2            Step 2This way, we have found that,   P  (  x  &amp;lt;  r  )  =            r      2              R      2      The probability that a dart falls within 1in of the center, is   P  (  x  &amp;lt;  1  i  n  )  =            (      1      i      n              )        2                    (      10      i      n              )        2              =      1    100  As the probability of throwing to the right or the left of the left has no effect on the distance to the center, this dooes not influence the result.

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