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Lesson 7 - The Elements of Probability When you walk into a room have you ever worried that all of the oxygen molecules will have moved to one corner, and that you will find it difficult to breathe? Chances are you have not (at least not before now). If you were asked to say why not, your answer would undoubtedly involve, in some way, the idea that this is a very unlikely, or improbable, event. (You might be tempted to say impossible, but that is not strictly true.) This is one example, albeit a rather silly one, of an informal application of the concepts of probability (and also a intrinsic understanding of molecular motion). You will encounter applications of probability throughout your study of chemistry (and other sciences). Since the quantum revolution of the early 20th century we know that the very laws of nature are based on probabilities rather than on the deterministic concepts that had been articulated by Newton and others. The first place that chemists have to learn to deal with these ideas is usually when talking about the structure of atoms, and the probabilistic response to the question "where is (are) the electron(s)". In your study of thermodynamics you will find that, in the absence of external forces, all chemical systems change in the direction of the state of maximum entropy, a state that we call equilibrium. That state is the most probable state for the chemical system under the particular conditions of temperature, pressure etc.. In fact, one definition of entropy, one that is a little more precise than just saying that entropy is a measure of disorder, is that of Boltzmann, who defined entropy mathematically in terms of the number of ways of achieving a particular state, W. His equation, S=k log W, is inscribed on his tombstone. (In Boltzmann's time, the expression log was used for natural logarithms, so today we write his equation as k ln W.) Our discussion/exploration of probability will begin on a seemingly more mundane level - tossing pennies and trying to win the Powerball game. Let's begin with penny tossing. A number of different aspects of probability are involved in this seemingly simple game. Let's start with the simplest. Whenever you perform an action such as tossing some coins, or making a measurement, the sum of the probabilities of all possible outcomes equals one. For the penny example, assuming a conventional penny with a "heads" side and a "tails" side, tossing the coin and allowing it to land on a flat surface has two possible outcomes (OK, it could land on its edge, but that is a very low probability event, and we will ignore it for this discussion), "heads" or "tails". The sum of the probabilities of those two outcomes equals one - this is another way of saying that you must get some outcome. This may seem a trivial point, but is important to keep in mind. Using our idealized penny, and avoiding for the moment discussions of differential air resistance on the two sides due to the artwork or imperfections in the coin, we can also assume that the two possible outcomes to tossing one penny have equal probabilities, and the probability of each must then be 0.5 (it actually makes more sense, as we will see, to write this as 1/2).
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