[CYO Butch]

Originally posted by Dan_ref

Sigh...if you wanna play dueling degrees I'm afraid you're not gonna come out too well. So let's just stick to the point, OK?

Probability does not apply to a *single* random test taker. This is a fundemental concept. A single random test taker could get all, some, or none right in a 100 question T/F test and his results can not be predicted by the laws of statistics. Also, as Mark said, for N large we should expect SOME guessers to get 100% correct. But for N large the guessers tend to converge on a score of 50% correct with a probability approaching 1.

And since you live in Nevada howzabout you put $100 on 00 for me, OK? Keep doing it until I win!

[Edited by Dan_ref on Jun 5th, 2003 at 10:23 PM]

Gotta side with Nevada on this. The laws of probability (which is really the relevant area) do apply equally to individuals and groups. If each "guess" is an independent event with two equally probable outcomes, it makes no difference whether the event is triggered by one or many different individuals. Thus, to continue with the coin toss analogy, the probability of 100 people each tossing a head with a single toss is the same as one individual tossing 100 out of 100 heads. The only issues that matter for the topic at hand are whether each "guess" is truly a random event (i.e., not influenced by prior events) and whether the two possible outcomes (T or F) are equally probable. Theoretically one could quantify those factors as well and develop a complete probability model to take them into account. My take is that they are not germain to Nevad's point that guessing should be penalized. The bottom line is that while I can never predict the outcome of a random event, I can assign a probility to it. I can also assign a probabilty to combinations and permutations of independent random events. I can give a probability for 7 out of 7 heads in a row, 7 out of ten, seven in a row in a series of ten tosses, etc. In fact, I can determine the exact probability of ANY possible outcome in the situation being described. Once actual results have been recorded, those results will fit into some place in the computed probability distribution. If the individual repeast the same series of questions randomly guessing again, his results will again fit into one of the slots. In fact, the distribution of these predicted outcomes fit a polynomial distribution which, at the size we are dealing with, is "normal" for all practical puposes. Over time, the frequency of his results will converge on the probabilities predicted. This is where statistics (as opposed to straightforward probability theory) comes into play.

Statistics are generally used when one is trying to extrapolate (predict) population characteristics or parameters from sample statistics. Under those situations, increases in the sample population will tend to produce data points from the sample that are normally distributed about the true value of the parameter of the underlying population. This is independent as to whether the characteristic iself follows a normal distribution in the actual population. (If I determine the average age of many different classes of 9th graders in New York, those averages will tend to be normally distributed around the actual mean of all 9th graders in New York. This is true even though the actual ages of the 9th graders are not likely to be normally distributed.) This principle can be used to predict the probability that the results from any given sample accurately reflect the underlying population. For example, it could be used modify the assumption that guessers on a test are equally likely to guess True and False. It also has the same basis as the statement that the more often an individual repeats the guessing process, the more likely his results will match the predicted outcomes. However, it has nothing to do with calculating the probability of the different possible outcomes from multiple coin tosses, or multiple random guesses between two possible answers.