The problem is as follows:
Six players draw, one after another and independently, a number uniformly distributed on $[0,1]$.
A player is called a recordist if he draws a number that is larger than all the numbers drawn by players before him.
What is the expected value of the number of recordists?
Attempted solution:
I define $X_i,\ 1 \leq i \leq 6$, as the indicator of the $i$th player being a recordist. In other words, $$X_i = \left\{ \begin{array}{ll} 1 &, \text{if the $i$th player is a recordist} \\ 0 &, \text{else} \end{array} \right. $$ Now since for all $\,i \neq j, \; \Pr(X_i \gt X_j ) = 0.5\,$, we have $$\Pr(X_i = k) = \left\{ \begin{array}{ll} \left(\frac{1}{2}\right)^{i-1} &, k=1\\ \ \ \ \ 0 &, k=0 \end{array} \right. $$ Next, $X = \sum_{i=1}^{6} X_i$ , is the number of recordists, and so $$ \mathbb{E} X = \mathbb{E} \left(\sum_{i=1}^{6} X_i\right) = \sum_{i=1}^{6} (\mathbb{E} X_i) = \sum_{i=1}^{6} \left(\frac{1}{2}\right)^{i-1} = \sum_{i=0}^{5} \left(\frac{1}{2}\right)^{i} = \frac{63}{32} $$ This is not the correct answer and so I turn to you for help. Thx, Gal.
If I understand your question correctly then the answer is $\frac {1764} {720} = \frac {49} {20}$.
To each person assign a ranking. The person who draws lowest has ranking 1, the person who draws the largest number has ranking 6.
List the rankings of a drawing in order as an array, $S$, by the order of draw; for example, one $S$ could be $\{3, 4, 1, 6, 5, 2\}$. The number of recordists is given by the length of a set: $$|\{i\, \text{ s.t. } (\forall\, j < i)\, S_j < S_i\}|$$
There are $6!$ possible permutations. With an epiphany we see that all permutations are equally likely. There is probably some clever way to divide the $720$ permutations into cases, for example we know that all permutations starting with $6$ have only $1$ recordist, the first drawer. You can probably recursively define the rest based on that property.
I found it easier to run the $720$ cases through a C program and got $1764$.
Calling the number of players $N$, and given that there are $N!$ permutations which forms the denominator for the probability, the numerator of this probability forms a sequence as a function of $N$ which is apparently something called "Unsigned Stirling numbers of the first kind". Here is its entry: Online integer Encyclopedia.
They don't show any nice closed form, which means using a computer was probably the only approach for a general $N$.