You will independently sample 10 times uniformly at random from the interval [0, 1]. For example, 0.767, 1/π , and 0.25 are all possible values you could find. At the end of this procedure, we will order the samples from least to greatest. Before sampling any numbers, I offer you the following proposition – I’ll pay you $100 if, after seeing only the first sample (call it x), you can guess what position x will eventually fall in the rankings. For example, if you guess that x will be 3rd smallest, and after seeing the rest of the numbers, this turns out to be true, then I’ll pay you $100. If I charge you to participate in this game, what is the maximum amount of money you would be willing to pay in exact dollars and cents?
Ranking Guessing Game: How Much Would You Pay to Predict the First Sample's Position?
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blamocur
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To answer such question I would first figure out an algorithm for making such guess. What are you thoughts?
blamocur
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Rank probabilities. A general question: given a random pick [imath]a[/imath] out of a sample of [imath]n[/imath] random numbers from the [0,1] interval what is the probability of its rank being [imath]m[/imath], where [imath]1\leq m \leq n[/imath]. For that we would have to have [imath]m-1[/imath] numbers below [imath]a[/imath] and [imath]n-m[/imath] numbers above it, and the corresponding probability is
[math]P_{m,n}(a) = {n-m\choose n-1} a^{m-1} (1-a)^{n-m} = {m-1\choose n-1} a^{m-1} (1-a)^{n-m}[/math]What is the best guess for [imath]m[/imath], i.e., which [imath]m[/imath] gets maximum for [imath]P_{m,n}[/imath]? As show elsewhere the maximum is achieved for [math]m= \lfloor na\rfloor[/math]Probability of correct guess. The average -- over all [imath]a[/imath]'s such that [imath]0\leq a \geq 1[/imath] -- value of [imath]P_{m,n}(a) = P_{\left\lfloor\frac{a}{10}\right\rfloor,10}(a)[/imath] is about 0.36602. Actual probabilities for different [imath]a[/imath]'s are plotted below:
[math]P_{m,n}(a) = {n-m\choose n-1} a^{m-1} (1-a)^{n-m} = {m-1\choose n-1} a^{m-1} (1-a)^{n-m}[/math]What is the best guess for [imath]m[/imath], i.e., which [imath]m[/imath] gets maximum for [imath]P_{m,n}[/imath]? As show elsewhere the maximum is achieved for [math]m= \lfloor na\rfloor[/math]Probability of correct guess. The average -- over all [imath]a[/imath]'s such that [imath]0\leq a \geq 1[/imath] -- value of [imath]P_{m,n}(a) = P_{\left\lfloor\frac{a}{10}\right\rfloor,10}(a)[/imath] is about 0.36602. Actual probabilities for different [imath]a[/imath]'s are plotted below: