-1
$\begingroup$

Let's say one wishes to place a bet on one of two fighters. You are given the following information:

(1) Fighters taller than their opponents by 3% will more likely win. (2) Fighters younger than their opponents by 3 or more years will more likely win.

You are also given this information:

Statement (1) is 58% accurate. Statement (2) is 60% accurate.

If Fighter1 is 4% taller than Fighter2, and Fighter2 is 4 years younger than Fighter1, who would you bet on, and why?

To be clear, I'm really interested in a more general case, where you have $n$ statements with $x_n$% accuracy. The above example is a bit easy. :)

Allow me to ask this question from another angle:

Let's say you have n people and each of them gives a prediction on who will win the fight between two fighters. Each person's prediction can be considered to be $x_n$% accurate based on their previous predictions. How would you consolidate all n predictions to get a more accurate prediction? Would you consolidate them at all, or just go with the best predictor? What if everyone disagreed with the best guy?

  • 2
    how is the above example easy? Taller and younger fighters are more likely to win, but *by how much*? Does 4% taller stand a better chance than 3% or is it constant past that threshold? The problem doesn't seem well posed as it's currently written.2012-07-12
  • 0
    Assume that it's constant past that threshold.2012-07-12
  • 1
    When you say Statement(1) is 58% accurate, do you mean that "fighters taller than their opponents by $\geq$3% win 58% of the time"? Because this is different than, say, "58% of taller fighters win 1% more often"2012-07-12
  • 0
    I mean that "fighters taller than their opponents by ≥3% win 58% of the time." In other words, out of 100 fights, 58 fights were won by opponents who were ≥3% taller.2012-07-12
  • 0
    The reason the above example is easy is this: there are only two statements and they are both applicable. However, one is more accurate. given that this is all the information available, you'd choose to bet on fighter2 since statement 2 is more likely to be right. It wouldn't be much of an edge but, again, the info is limited.2012-07-12
  • 3
    By that way of counting, the statement _"after getting heads once, the next flip of a fair coin is more likely to be tails"_ would be considered 50% accurate. I would rather call it 0% accurate.2012-07-12

1 Answers 1

2

We don't have enough information. Let's say for simplicity we divide the universe of fighters into four groups: 1 (short and old), 2 (short and young), 3 (tall and old), 4 (tall and young), and $p_{ij}$ is the probability of a fighter from group $i$ winning against a fighter from group $j$. We also should know what fraction of all fights involve a fighter from group $i$ against a fighter from group $j$: say $w_{ij}$ (where $w_{ij} = w_{ji}$ and $\sum_{i \le j} w_{ij} = 1$). Then statement (1) says $p_{31} w_{31} + p_{32} w_{32} + p_{41} w_{41} + p_{42} w_{42} = 0.58 (w_{31} + w_{32} + w_{41} + w_{42})$, and statement (2) says $p_{21} w_{21} + p_{23} w_{23} + p_{41} w_{41} + p_{43} w_{43} = 0.6 (w_{21} + w_{23} + w_{41} +w_{43})$. But those are just two equations in six unknowns.

  • 0
    You're absolutely right. There is not enough information. But there is some information! Imagine you had a gun to your head. :) Which would you choose?2012-07-12
  • 2
    @Korgan: I am thinking of a number between 1 and 10. Imagine you had a gun to your head. What number am I thinking of?2012-07-13
  • 0
    You have a kid who very often drops plates. Give him a plate. Will he drop it? The answer is probably/more likely than not, in which case you'd bet on the kid dropping the plate. How likely? Who cares. That wasn't my question. If you're thinking of a number, I have no idea. I have no other information. If I had your history of number-picking, I may or may not have an advantage with my guess.2012-07-13