Question 1210189
.
In a basketball tournament, there are four teams, and each team plays against every other team exactly twice.  
(So each team plays six games.  Also, each team is equally likely to win a game, and there are no ties.)  
Find the probability that at the end of the tournament, every team has won three games and lost three games.
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        In the post by @CPhill the solution is not mathematical:  it is obtained by using numerical modeling

        with a computer.   An adequacy/(a correctness)  of the computer code and of this solution is unclear and is unknown.


        In this my post,  I will give a pure mathematical solution, as it is expected.

        It gives another answer, distinct of that by @CPhill.



<pre>
For each game, the probability to win/lose is 1/2.


So, for each team, playing with other teams is a binomial experiment
with n = 6 (the number of trials) and p = 1/2 (the probability for each individual win/loss).


Hence, for each team, to win 3 and to lose 3 games has the probability

    P = {{{C[6]^3*(1/2)^3*(1/2)^3}}} = {{{20/64}}} = {{{5/16}}}.


For all four teams, it is intuitively clear that the results of its playing of one team with the other teams
are independent from the results of playing of the second team with the other teams.


(Actually, it is a general property of any binomial distribution).


Therefore, the probability of getting 3 win and 3 lose in this problem is  {{{(5/16)^4}}} = {{{625/65536}}} = 0.009536743 (approximately).


<U>ANSWER</U>.  In this problem, the probability that every team has won three games and lost three games is  

         {{{(5/16)^4}}} = {{{625/65536}}} = 0.009536743 (approximately).
</pre>

Solved.


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As I just said at the beginning of my post, my result/answer is different from that by @CPhill.


Interesting that &nbsp;Google &nbsp;AI &nbsp;gives the same answer as my solution 
(although their method of calculations is different from mine).


Regarding this &nbsp;Google &nbsp;AI &nbsp;solution, &nbsp;look at the link


https://www.google.com/search?q=In+a+basketball+tournament%2C+there+are+four+teams%2C+and+each+team+plays+against+every+other+team+exactly+twice.+(So+each+team+plays+six+games.+Also%2C+each+team+is+equally+likely+to+win+a+game%2C+and+there+are+no+ties.)+Find+the+probability+that+at+the+end+of+the+tournament%2C+every+team+has+won+three+games+and+lost+three+game&rlz=1C1CHBF_enUS1071US1071&oq=In+a+basketball+tournament%2C+there+are+four+teams%2C+and+each+team+plays+against+every+other+team+exactly+twice.+++(So+each+team+plays+six+games.++Also%2C+each+team+is+equally+likely+to+win+a+game%2C+and+there+are+no+ties.)+++Find+the+probability+that+at+the+end+of+the+tournament%2C+every+team+has+won+three+games+and+lost+three+game&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTIwMDdqMGoxNagCCLACAfEF4pOwJLJVWGU&sourceid=chrome&ie=UTF-8