How Do You Know That Events Are Equally Likely?
Date: 07/03/2008 at 15:19:57 From: Chris Subject: "equally likely" in regards to probability How do you determine whether the events of a problem are equally likely? I can't seem to find any information regarding how to determine if events are equally likely. Most texts don't get into it at all and others use circular logic to show it. For example, the events of a fair coin toss are equally likely because they each have a probability of 1/2. But you can only use that calculation once you have determined that the events are equally likely. How do you make that determination?
Date: 07/03/2008 at 23:01:15 From: Doctor Peterson Subject: Re: "equally likely" in regards to probability Hi, Chris. This is an excellent question! There are basically two answers, neither of which will sound right until we put them in context. Make that three, actually, since we'll want to talk about the practical matter of doing your homework, too. The first answer is that we don't KNOW at all; we ASSUME. You may have noticed that in many math problems they will say "Assume the coin is fair", or something like that. This may seem like cheating, but it is really what math is all about: reasoning from "axioms" (basic assumptions) that define the subject of our reasoning. Whether the assumption makes sense is the subject of the second answer, but we're not ready for that yet! In math, we commonly start with a model of some concept in the world, such as a fair coin or a flat surface; rather than deal with all the complexities of real coins or surfaces, we think about what would be true of an IDEAL coin (heads and tails are equally likely, and there's no other option like standing on edge) or plane (you can draw one line through any two points, etc.). Then we reason based on those assumptions, so that all our conclusions will be definitely true IF those assumptions are true. The second answer is that we don't know anything PERFECTLY; we APPROXIMATE. Any real coin is likely to be a little biased in one direction or the other; and there will not be exactly as many boys born as girls. But experience tells us that it is reasonable to assume, for many purposes, that an ordinary coin is likely to be very close to fair, and that one is about as likely to have a boy as a girl. So we make those "simplifying assumptions" when we don't need extreme precision in our answers. Sometimes we do want to be really sure of our answers (in the real world--maybe because real money depends on it); then we don't just go by general EXPERIENCE, but by careful EXPERIMENT. We toss a thousand coins thousands of times each under controlled conditions and determine just how close to fair the average coin is, and how far from fair any given coin is likely to be. This field of study is called statistics, and it can provide the basis for exact calculations of real probabilities--as far as we know, and as long as the population of coins or children we are working with matches the one we studied. Again, we can never be exactly sure ... (By the way, studies of tossed coins have shown that while actually tossing a penny in the air is quite fair, spinning one on the table is not; the head side is slightly heavier, and it will fall heads down twice as often as heads up! So the assumption of fairness would not be true if the coin rolls on a surface before falling.) The third answer is that in problems you are given, you are expected to choose equally likely outcomes based on a combination of standard assumptions that have been presented in your text or elsewhere, common sense, and your knowledge of probability. For example, when you toss two coins, you either have been told to assume they are fair coins, or you know from experience or from statistics that they are close to fair, so it makes sense to consider heads and tails on EACH INDIVIDUAL coin as equally likely. You also know enough about probability to realize that that assumption would conflict with the easier assumption that "no heads", "one head", and "two heads" are equally likely. In particular, you have learned that compound events such as this can't be assumed to be equally likely, but simple events (like a single coin) often can. This becomes a sort of intuition: you've seen it happen enough that you will be much more willing to assume that simple events are equally likely than that more complicated things are. You break things down to the simplest possible parts, and then decide whether it makes sense to suppose that those are equally likely. So here's my answer: Math is not directly about the real world, but about simplified models of the real world. When we are working within math itself, just practicing or developing its techniques, we are happy just to make assumptions, which merely define what we are talking about. When we are working with something real, such as gambling at a casino, we have to determine whether our model is realistic, so we use some sort of statistics to check whatever assumptions we make. (Professional dice are made with extreme care and tested to ensure that they are extremely fair, so that one can make the "simplifying assumption" with confidence. But if someone were to roll double sixes 100 times in a row, you would have reason to question the assumption, and ask to see the dice!) Finally, when you are doing problems in a math class, if assumptions are not explicitly stated, you can generally just use common sense as long as you are dealing with simple outcomes. I should note one other thing: It is possible to solve problems WITHOUT any equally likely outcomes, and higher level study of probability does just that. The idea of equally likely outcomes just makes it easy to explain the basic concepts, and to solve problems. If you are told that a coin had a probability of 0.30 of heads and 0.70 of tails (as for that spinning penny), you can just take THAT as your assumption, with no actual equiprobable events in sight. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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