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[SOLVED] Probability of Pairings in Chess Game
The chess clubs of two schools consist of, respectively, 8 and 9 players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing plays a game of chess. Suppose that Rebecca and her sister Elise are on the chess clubs at different schools. What is the probability that
(a) Rebecca and Elise will be paired;
(b) Rebecca and Elise will be chosen to represent their schools but will not play each other;
(c) exactly one of Rebecca and Elise will be chosen to represent her school?
Axioms and basic theorems of probabilitiy.
(a) Assuming each possible team is equally likely and each possible pairing is equally likely, the the probability sought is the ratio of the number of possible teams and pairing in which Rebacca and Elise are paired to the total possible number of teams and pairings which should be
\frac{\binom{7}{3}\binom{8}{3}3!}{\binom{8}{4}\binom{9}{4}4!} = \frac{1}{14}
The book says the answer is 1/18 however.
(b) That should be
\frac{\binom{7}{3}\binom{8}{3}3 \cdot 3!}{\binom{8}{4}\binom{9}{4}4!} = \frac{3}{14}
The book says the answer is 3/18 however.
(c) That should be
\frac{\binom{7}{3}\binom{8}{4}4! + \binom{7}{4}\binom{8}{3}4!}{\binom{8}{4}\binom{9}{4}4!} = \frac{1}{2}
The book has the same answer.
My gripe is with (a) and (b). Is the book right or am I right?
Homework Statement
The chess clubs of two schools consist of, respectively, 8 and 9 players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing plays a game of chess. Suppose that Rebecca and her sister Elise are on the chess clubs at different schools. What is the probability that
(a) Rebecca and Elise will be paired;
(b) Rebecca and Elise will be chosen to represent their schools but will not play each other;
(c) exactly one of Rebecca and Elise will be chosen to represent her school?
Homework Equations
Axioms and basic theorems of probabilitiy.
The Attempt at a Solution
(a) Assuming each possible team is equally likely and each possible pairing is equally likely, the the probability sought is the ratio of the number of possible teams and pairing in which Rebacca and Elise are paired to the total possible number of teams and pairings which should be
\frac{\binom{7}{3}\binom{8}{3}3!}{\binom{8}{4}\binom{9}{4}4!} = \frac{1}{14}
The book says the answer is 1/18 however.
(b) That should be
\frac{\binom{7}{3}\binom{8}{3}3 \cdot 3!}{\binom{8}{4}\binom{9}{4}4!} = \frac{3}{14}
The book says the answer is 3/18 however.
(c) That should be
\frac{\binom{7}{3}\binom{8}{4}4! + \binom{7}{4}\binom{8}{3}4!}{\binom{8}{4}\binom{9}{4}4!} = \frac{1}{2}
The book has the same answer.
My gripe is with (a) and (b). Is the book right or am I right?