Results 1 to 5 of 5
Like Tree2Thanks
  • 1 Post By romsek
  • 1 Post By romsek

Thread: probability problem about tournaments

  1. #1
    Newbie
    Joined
    Oct 2017
    From
    Hong Kong
    Posts
    10

    probability problem about tournaments

    A, B and C are three friends who often like to compete with one another.
    (a) they are equally skillful in playing table tennis. A plays B first while C waits. the winner then plays C while the loser waits. This method of play goes on until one of them wins two consecutive sets. this person then gets a prize. find the probability that C gets the prize. ( ans: 2/7)
    (b) they are given the same multiple-choice quiz to answer independently. the quiz has 10 questions, each provided with 4 possible answers. only one of which is correct. A knows eight of the correct answers, B knows seven, and C knows six. All of them then answer the remaining questions by sheer guess. No penalty is imposed on a wrong answer. What is the probability that C beats A and B?
    (ans: 0.0259)

    I can't figure out the method of solving this problem even after looking at the answers. Can someone help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,880
    Thanks
    2473

    Re: probability problem about tournaments

    For (b)

    List out the combinations of successful guesses that allow C to beat A and B. We get

    $\begin{matrix}A &B &C \\--- &--- &--- \\
    0 &0 &3 \\
    0 &0 &4 \\
    0 &1 &3 \\
    0 &1 &4 \\
    0 &2 &4 \\
    1 &0 &4 \\
    1 &1 &4 \\
    1 &2 &4
    \end{matrix}$

    A is a binomial distribution $Binomial(2,1/4)$
    B is $Binomial(3,1/4)$
    C is $Binomial(4,1/4)$

    It's assumed the guessing is independent amongst the students so the probability of each row of the table is just the product of the 3 individual probabilities.

    Then sum up all the row probabilities to get the final probability that C beats A and B

    probability problem about tournaments-clipboard01.jpg
    Thanks from elmomleo
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    5,880
    Thanks
    2473

    Re: probability problem about tournaments

    a)

    Look at this state diagram

    probability problem about tournaments-clipboard01.jpg

    Two things to note.

    i) The left side, which I didn't draw, is identical in probability to the right side. So each side will contribute half of the overall probability.

    ii) Each vertex has probability $\dfrac 1 2$ as all the players are equally skilled.

    If you examine the diagram you'll see that C wins at $p=(1/2)^3$, and $p=(1/2)^6$ and will continue to win at $p=(1/2)^{3k}$

    so the total probability of C winning is

    $\begin{align*}
    &P = 2 \sum \limits_{k=1}^\infty~\left(\dfrac 1 2 \right)^{3k} = \\
    &2 \sum \limits_{k=1}^\infty~\left(\dfrac 1 8\right)^{k} = \\
    &2\left(\dfrac {1}{1-\frac 1 8}-1\right) =\\
    &2\left(\dfrac 8 7 - 1\right) = \\
    &\dfrac 2 7
    \end{align*}$
    Thanks from elmomleo
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2017
    From
    Hong Kong
    Posts
    10

    Re: probability problem about tournaments

    So two consective sets means winning A and B continuously?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2010
    Posts
    2,971
    Thanks
    1141

    Re: probability problem about tournaments

    It means C wins twice in a row. Not continuously, but consecutively.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 'strong' tournaments
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Apr 26th 2017, 07:22 PM
  2. Replies: 10
    Last Post: Jan 21st 2011, 12:47 PM
  3. Replies: 0
    Last Post: Oct 8th 2009, 09:45 AM
  4. recurrence relations tournaments
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: May 2nd 2009, 04:25 PM
  5. A PROBLEM in PROBABILITY
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: Apr 13th 2009, 07:21 PM

Search Tags


/mathhelpforum @mathhelpforum