Explore BrainMass

Explore BrainMass

    Probability : Games Won

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    1. Suppose that two teams play a series of games that ends when one of the teams has one i number of games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when (a) i = 2 and when (b) i = 3. Show also in both cases that this number is maximized when p = ½.

    2. Consider problem 1 with i = 2. Find the variance of the number of games played and show that this number is maximized when p = ½.

    © BrainMass Inc. brainmass.com March 4, 2021, 6:13 pm ad1c9bdddf
    https://brainmass.com/math/probability/probability-games-won-35482

    Attachments

    Solution Preview

    The two teams play a series of games that ends when one of them has won i games
    P(A) = p, then P(B) = 1-p

    (a) when i = 2, we need at the most 3 rounds.
    In the first round, the probability of A winning is p
    In the second round, the probability of A winning again is p, when A won the whole game.
    Then probability for A to finish the game in two rounds is p*p = p^2
    ("^" is the power of)
    However, the same story applies to team B, probability for B to finish the game in two rounds is (1-p)^2
    the total probability of finishing the game in two rounds is p^2+(1-p)^2

    If the game goes to the third round, the probability is then 1-( p^2+(1-p)^2)
    Then the expected number of games (rounds) that are played is
    N = 2*( p^2+(1-p)^2) + 3*(1-( p^2+(1-p)^2))
    = 3 - (p^2+(1-p)^2)
    = 3 - (2 p^2 - 2p +1)

    To maximize N, we write:
    N = 3 - (2 p^2 - 2p +1)
    = 3 -2 (p^2 - p +0.5)
    = 3 -2 (p^2 - p +0.25) - 0.5
    = 2.5 -2 (p- 0.5)^2

    To maximize N, we need p-0.5=0, then p = 0.5
    And N(max) = 2.5 - 0 =2.5

    [ also we can deduct the following caculation:
    when i = 2, we need at the most 3 rounds.
    In the first round, the probability of A winning is 0.5
    In the second round, the ...

    Solution Summary

    Probability of games won is investigated. The solution is detailed and well presented.

    $2.49

    ADVERTISEMENT