How to find the mathematical expectation?

Question

$190$ people take part in the marathon, and $124$ of them use the youth team of the university. The random value $\zeta$ is equal to the place of the best member of the university team. How to find the mathematical expectation of $\zeta$?

Answer 1

Of the competitors $66$ are not members of the university team.

For $i=1,\dots,66$ let random variable $X_{i}$ take value $1$ if person $i$ gets a higher ranking then all members of the university team and let $X_{i}$ take value $0$ otherwise.

Then: $$\zeta=1+\sum_{i=1}^{66}X_{i}$$

Now apply linearity of expectation and symmetry.

Answer 2

Although one may prefer drhab's solution, here a direct calculation:

How many possibilities are there to place 124 students to 190 positions?

Obiously $\binom{190}{124}$

1: How many possibilities are there that the university wins the 1st price? So: One student on the first place, the rest we can place randomly on the other places… but we have $\binom{189}{123}$ possibilities for this

So the chance to win the first price is $$\frac{\binom{189}{123}}{\binom{190}{124}} = \frac{124}{190}$$

2: How many possibilities are there that the university wins the 1st price? So: So no student on the first place, one student on the second, the rest we can place randomly on the other places… but we have $\binom{188}{123}$ possibilities for this. So the chance to win the second price is $$\frac{\binom{188}{123}}{\binom{190}{124}}$$

$\vdots$

67: How many possibilities are there that the university wins the 67st price? So: So no student on the first 66 places, one student on the 67, the rest we can place randomly on the other places… but we have $\binom{123}{123}$ possibilities for this. So the chance to win the second price is $$\frac{\binom{123}{123}}{\binom{190}{124}}$$

Denote $X$ the place of the university then we get: $$E[X] = \sum_{j=1}^{67} jP(X=j) = \frac{1}{\binom{190}{124}} \sum_{j=1}^{67} j\binom{190-j}{123} = \frac{191}{125} = 1.528$$ Whereas the last calculation you can directly calculate e.g. using WolframAlpha