How to calculate the expected average rally length in tennis

Discussion in 'Physics & Math' started by Jennifer Murphy, Jun 20, 2019.

Thread Status:
Not open for further replies.
  1. Jennifer Murphy Registered Senior Member

    Messages:
    239
    I would like to calculate the expected average rally length (number of returns before an error) in tennis given the probability that both players will return the ball.

    Let A = the probability that Player A will return the ball and B = the probability that Player B will return the ball. Then (1-A) = probability that A will miss and (1-B) = probability that B will miss.

    It seems to me that the expected average rally length should be the sum of the counting numbers (1, 2, 3, ...) multiplied by the probability that they will occur. Therefore, the series should be:

    0*(1-A) + 1*A*(1-B) + 2*A*B*(1-A) + 3*A*B*A*(1-B) + . . .

    Is that correct? It seems like is should be, but I keep getting odd results with it.
     
  2. Google AdSense Guest Advertisement



    to hide all adverts.
  3. mathman Valued Senior Member

    Messages:
    2,002
    You need to give some details for the odd result. Did you verify that the probabilities add up to 1?
     
  4. Google AdSense Guest Advertisement



    to hide all adverts.
  5. Jennifer Murphy Registered Senior Member

    Messages:
    239
    Part of a more complicated simulation calculates the average rally length and those numbers do not agree with the result of the formula above. I don't have a closed form fopr the infinite series, so I can't verify it exactly, but I wrote a little program to calculate 100 terms and that does not agree.

    But what difference does that make? The formula above is either right or it isn't. Do you know which it is?

    If you mean the probabilities assigned to the counting numbers above, then I think they do add up to 1.

    Stripping off the "coefficients" (counting numbers), we have,

    (1-A) + A(1-B) + AB(1-A) + ABA(1-B) + ABAB(1-A) + ABABA(1-B) + . . .

    = 1 - A + A - AB + AB - A^2B + A^2B - A^2B^2 + A^2B^2 -A^3B^2 + A^3B^2 - A^3B^3 + . . .

    All the terms but the first and last cancel out and the last, A^infinityB^infinity, should be zero.

    Correct?
     
  6. Google AdSense Guest Advertisement



    to hide all adverts.
  7. James R Just this guy, you know? Staff Member

    Messages:
    39,397
Thread Status:
Not open for further replies.

Share This Page