Math Help? Probability?
The following table shows the results of six football games between the teams A, B, C and D, in which each team played any other team. What were the scores of every game? Scored for each of the two teams play list and the number of goals, each team. Diagram here: http://i358.photobucket.com/albums/oo24/xphilophobia/math-1.jpg Show or explain your approach and your reply.
Let the games be ordered so that 1 = A plays B game, the score as A1 and B1 Game 2 C = A plays, the score as A2 and C2 = A plays Game 3 D. … Game 4 = B plays C = B plays Game 5 Game 6 D = C plays D, the "goals for" column defines four Equations A1 + A2 + A3 = 7 B1 + B4 + B5 = 1 C2 + C4 + C6 = 3 D3 + D5 + D6 = 3, the "goals against" column defines B1 + C2 + D3 = 0 A1 + C4 + D5 = 1 A2 + B4 + B5 + A3 = 8 D6 + C6 = 5 Now, unfortunately, we have 12 unknown variables, and only eight equations. Fortunately, because B1 + C2 + D3 = 0, and scores can not be negative, We know, B1, C2 and D3 are all zero by effectively two new equations. We also know some disparities, such as A1> B1, A2> A3 and C2> D3, because A has won all his Games. But that tells us not only the A1 is zero. Since C tied two games, they must have bonded with B and D. This means, B4, and C6 = C4 = D6. We now have enough equations to solve. The simplification of the equations A1 + A2 + A3 + B5 = C4 = 7 1 + 3 = C4 D6 D5 + D6 = A1 3 + C4 + D5 = C4 + 1 A2 + A3 + B5 = 8 D6 + D6 = 5 Now, remember, that A1 is not zero, it must be 1, forcing C4 and D5 to zero. Simplifying again A2 + A3 = 6 (since 1 = 1) C4 + B5 = C4 + 1 = 3 D6 D5 + D6 = 3 A1 = 1 A2 + A3 = 8 D6 + B5 + D6 = 5 The next step is to recognize that C4 and B5 are forced to be zero or 1. Letting C4 = 1 leads to a contradiction (this solution), but B5 no = 1. C4 = B5 = 1 A2 = 0 5 = 3 D5 D6 = 0 A2 = 5 A3 = 1, so that the grades are A1 = 1, B1 = 0 Taken A, B Loser of A2 = 5, C2 = 0 Taken A, C Loser A3 = 1, D3 = 0 A winners, losers B4 D = 0, C4 = 0 B tie, tie B5 C = 1, D5 = 0 B winners, losers D C6 = 3, D 6 = 3 C tie, tie D Checking A won all three games B won one game, C & D is never won. Tally scores! Good luck.
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