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1994 Paper 1 Q12
D: 1500.0 B: 1468.0
probability conditional probability combinatorics counting college selection Bayes' theorem

There are 28 colleges in Cambridge, of which two (New Hall and Newnham) are for women only; the others admit both men and women. Seven women, Anya, Betty, Celia, Doreen, Emily, Fariza and Georgina, are all applying to Cambridge. Each has picked three colleges at random to enter on her application form.

  1. What is the probability that Anya's first choice college is single-sex?
  2. What is the probability that Betty has picked Newnham?
  3. What is the probability that Celia has picked at least one single-sex college?
  4. Doreen's first choice is Newnham. What is the probability that one of her other two choices is New Hall?
  5. Emily has picked Newnham. What is the probability that she has also picked New Hall?
  6. Fariza's first choice college is single-sex. What is the probability that she has also chosen the other single-sex college?
  7. One of Georgina's choices is a single-sex college. What is the probability that she has also picked the other single-sex college?

Solution:

  1. \(\frac{2}{28} = \frac{1}{14}\)
  2. \(1-\frac{27}{28}\frac{26}{27}\frac{25}{26} = \frac{3}{28}\)
  3. \(1 - \frac{26}{28}\frac{25}{27}\frac{24}{26} = \frac{13}{63}\)
  4. \(1-\frac{26}{27}\frac{25}{26} = \frac{2}{27}\)
  5. \(\frac{1}{27}\)
  6. There are \(\binom{2}{1} \binom{26}{2} + \binom{2}{2}\binom{26}{1}\) ways to choose at least one single sex college and \( \binom{2}{2}\binom{26}{1}\) ways to choose both, therefore \begin{align*} P &= \frac{ \binom{2}{2}\binom{26}{1}}{\binom21 \binom{26}2+ \binom{2}{2}\binom{26}{1}} \\ &= \frac{26}{2\cdot \frac{26\cdot25}{2}+26 }\\ &= \frac{1}{25+1} = \frac{1}{26} \end{align*}

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