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PROFESSOR: Examine every possible means for combining the symmetry at once, but there is a seemingly paradoxical trick that we can yet pull. And let me indicate what is true here for 2mm. OK, so this is a twofold axis. Has mirror planes perpendicular to it. If one of these is a mirror plane, the other one has to be a mirror plane as well. So there's no way we could make one a mirror plane and one a glide plane. OK, that requires a net that is exactly rectangular. So let's put in the twofold axis. And I add one to the corner of the cell. As we well know we have to have twofold axes at all of these other locations. We want to put a mirror plane in the cell. We could pass it through the twofold axis, and that would be the same as P getting to P2mn back again. But why do we have to put the mirror plane through the twofold axis? We have to have the twofold axis left unchanged when we add the mirror plane, because if we created a new twofold axis we create a new lattice...

Let us now examine what conditional probabilities are good for. We have already discussed that they are used to revise a model when we get new information, but there is another way in which they arise. We can use conditional probabilities to build a multi-stage model of a probabilistic experiment. We will illustrate this through an example involving the detection of an object up in the sky by a radar. We will keep our example very simple. On the other hand, it turns out to have all the basic elements of a real-world model. So, we are looking up in the sky, and either there's an airplane flying up there or not. Let us call Event A the event that an airplane is indeed flying up there, and we have two possibilities. Either Event A occurs, or the complement of A occurs, in which case nothing is flying up there. At this point, we can also assign some probabilities to these two possibilities. Let us say that through prior experience, perhaps, or some other knowledge, we kno...

Let us examine how to interpret the model we developed. One of the things we should look after is that there might be what is called multicollinearity. Multicollinearity occurs when the various independent variables are correlated, and this might confuse the coefficients-- the betas-- in the model. So tests to address that involve checking the correlations of independent variables. If they are excessively high, this would mean that there might be multicollinearity, and you have to potentially revisit the model, as well as whether the signs of the coefficients make sense. Is the coefficient beta positive or negative? If it agrees with intuition, then multicollinearity has not been a problem, but if intuition suggests a different sign, this might be a sign of multicollinearity. The next important element is significance. So how do we interpret the results, and how do we understand whether we have a good model or not? For that purpose, let's take a look at what is called...