More options Jul 16, 1:24 pm Newsgroups: sci.math From: gearhead Date: Fri, 16 Jul 2010 10:24:41 -0700 (PDT) Local: Fri, Jul 16 2010 1:24 pm Subject: stats/probability question Reply | Reply to author | Forward | Print | Individual message | Show original | Remove | Report this message | Find messages by this author
I'm an engineering undergrad in an intro stats course. We had a question in the book that's really dumb.
problem as stated:
Your candidate has 55% of the votes in the entire school. But only
100 students will show up to vote. What is the probability that the underdog (the one with 45% support) will win? To find out, set up a simulation. a) Describe how you will simulate a component and its outcomes. b) Describe how you will simulate a trial. c) Describe the response variable.The answer in the back of the book says using a two digit random number to determine each vote (00-54 for your candidate, 55-99 for the underdog) you would run a string of trials with 100 votes to each trial.
Now, this is one misconceived exercise. Let me explain why.
Say the school has 1000 students. If all of them show up, the underdog has 0% chance of winning. If exactly one voter shows up, underdog has 45% chance of winning. In an election where 100 voters show up, underdog's chance of winning the election HAS to lie somewhere between 0% and 45%. No ifs, ands or buts. The probability of a win for underdog can never exceed 45%. When the exercise asks "how often will the underdog win" I interpret that as meaning what are his chances, i.e., the probability that he will win. But if you run a simulation, you can get anything, including results above 45%. I don't think simulating has any validity here, at least the procedure suggested in the answer key. That is a lot of simulating to do by hand, 100 per trial, but it is nowhere close to even starting to answer the actual question. You would first of all have to know the population of the school and then do some very demanding simulations that would only be practical on a computer.
Practical considerations aside, the question is meaningless unless know something about the magnitude of the school population. Consider: if the total population is 108, the underdog cannot win, because he only has 49 (48.6 rounded up) supporters total. Chance of winning 0%. Period. "Underdog" has NO CHANCE of winning the election. But if you run a simulation the way the book suggests, he's going to win some. In fact he wins about half. I'm saying the book is wrong. Back to our school of 1000 students, out of whom 450 would vote for "underdog." If only 100 students vote, what are his chances of winning? Simulation will send you on the wrong track here unless you're ready for some head scratching and a big grind on the computer. I'm sure this problem has a neat theoretical solution.
In class today I saw this problem and just was mystified until I worked out the implications, and now it's clear that it's just incredibly stupid. How would you convince the teacher of that? If I point out that it's impossible to get any answer above 45%, she might say, well this isn't theoretical, we're just running simulations, which is the whole point of the game. To convince her I might have to work out the actual correct simulation methodology, which is likely a very big headache and something I don't have time for. So I may just let it slide and not even bring it up. But I'm still interested in the theoretical solution, if anybody can cough it up. It's a probability problem now, not empirical statistics.
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