Quote:
Originally posted by Dan_ref
The real problem with his stance is that as a theory it is incorrect. Theories of random probability do not apply to any single person, they only apply to very large groups of people. In theory it is very possible for a person to guess on every T/F question and get 100% correct. As the number of people taking the test increases the probability that the overall score is 50% approaches 100%. Gambling casinos show these theories to be true every day.
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Dan,
First, I have a math degree. Second, I live in Nevada. So if you want to talk probability and gambling, I'm your man.
Unfortunately, you are not right in saying that my stance is incorrect. Here's why:
What you said above about the score of a single person taking the test has merit, but your claim is terribly misleading.
You are saying that if we give a bunch of officials only the answer sheet and let them just guess T or F, that their average score will approach 50% as the number of officials that we do this with gets bigger and bigger. That is absolutely true. However, you are also saying that if we select only one of these officials at random and look at his score that his score could be very far from 50%. You even used 100% as your example. This is the part of your claim that is misleading. While this is theoretically possible, it is HIGHLY UNLIKELY. This is due to the normal distribution of the "Bell" curve and the fact that the vast majority of scores are within a standard deviation or two of the mean.
Doing this is exactly the same as flipping a coin 100 times and recording the result. The official now uses Heads = true and Tails = false to fill in his test answers. The chance that someone could score 100% on the test doing this is one-half to the 100th power. That number microscopic! However, above you said, "In theory it is very possible for a person to guess on every T/F question and get 100% correct." Sorry, but no. It is certainly not "very possible." It is in fact, ALMOST IMPOSSIBLE. (This is why casinos make so much money. Ever play roulette?) Actually, there is the same probability that the official will miss all the questions and score a ZERO. These outlier values occur with such small frequency that we can basically ignore them. That is the whole basis of the theory of probability.
So all of this supports my claim that an official should expect to get roughly half of his guesses right on the test. Yes, some officials will get more than half right and others will get fewer, but it won't vary that much from the 50% figure.
For an example, with some hard numbers to back me up, if you give the test to 256 officials and each of them guesses on exactly 8 questions, you would only expect 1 official to get all 8 of his guesses correct. If each official guessed on 10 questions, you would have to give the test to 1024 people in order to expect a single person to get all 10 guesses correct. For 30 guesses, that number jumps to 1073741824. How many officials do you think we need to test before someone would correctly guess all 100 questions? The answer is 2 to the 100th power. A huge number.
Now remember, my problem is not with those officials who need to guess on only 10 questions (due to wording problems and some technical definitions most of us have a few guesses), but those who are so lacking in rules knowledge that they have to guess on about half of the test.
To get a hard number that we can work with let's say 50. If an official is guessing on fifty questions, his chance of nailing all of them is 1 in 2^50 (which I believe is a 16 digit number). However, he has about a 50% chance of getting either 23,24,25,26,or 27 of those guesses right and an 88% chance of scoring between 20 and 30 correct. So he can basically add a safe +20 to his score by guessing. Therefore, someone who knows only 50% of the stuff will look like he knows over 70%. That is not helpful for evaluating an official.
Therefore, the scoring of the test is misleading because the scores not only don't reflect your officiating ability, they don't even accurately approximate your rules knowledge!
Mick, that is what I object to. I believe that if we are going to go through the process of taking the test that we should at least make it meaningful in some manner.