The top students in a large high school graduation class are ranked from $1$ (best) to $10$ (tenth best). Assume that each rank is equally likely to be assigned to a male student or a female student. Let $X$ equal the sum of the ranks (from $1$ to $10$) that are assigned to female students, that is if all of the top $10$ students are girls, $X=1+2+3+...+10=55$. Provide the probability function on the sample space.

I believe that the total possible rankings is $2^{10} = 1024$, and I think that $P(X=k)$ is given by the ratio of (ways to get a sum $k$ of ranks assigned to females)/(all possible arrangements). I do not know how to represent the numerator as some type of function.

  • $\begingroup$ Thank you, Leucippus, for the edit. $\endgroup$ – Brian Wynne Sep 15 '18 at 1:46
  • $\begingroup$ Are you looking for a closed form formula or just the numbers in this specific case? $\endgroup$ – pepster Sep 15 '18 at 2:40

Here is a recursive Python function to compute the number of ways to get the sum total using 1,2,...,k

def f(total, k) :
  """ Number of ways to obtain 'total' as a sum of uniqe integers from 1..k """
  if total < 0 or k <= 0:
    return 0
  if 0 <= total <= 1:
    return 1

  if k > total:
    k = total

  return f(total, k-1) + f(total-k, k-1)

so $P(X=k) = \frac{f(k,n)}{2^n}$ ($n=10$ in your case). This kind of formula can be a good first step to validate any proposed closed form formulas.

  • $\begingroup$ This was most helpful, pepster, thank you. $\endgroup$ – Brian Wynne Sep 17 '18 at 1:30

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