Consider a purple urn containing 7 red marbles and 3 blue marbles as well as a green urn containing 5 blue marbles and 4 yellow marbles. An experiment consists of drawing a marble from the purple urn, placing it in the green urn, and then drawing a marble from the green urn. Given that the marble drawn from the green urn was blue, what is the probability that the marble drawn from the purple urn was red? Give your answer to three decimal places.

2. [Uncertainty and risk] A DM is presented with two jars. Jar 1 has 50 red and 50 blue balls. Jar 2 consists of 100 total balls each of which is either red or blue but the colors are in an unknown proportion. An experiment consists of drawing a single ball from each jar. The DM faces the following two choices. Choice 1 is between option 1a which pays $100 if the Jar 1 ball is red, and option 1b which pays $100 if the Jar 2 ball is red (and $0 otherwise). Choice 2 is between option 2a which pays $100 if the Jar 1 ball is blue, and option 2b which pays $100 if the Jar 2 ball is blue (and $0 otherwise). Suppose the DM chooses 1a over 1b and 2a over 2b (and has a strict preference in each case).

Are these choices consistent with subjective EU? In other words, does there exist a probability distribution over the contents of Jar 2 (that is, a belief that the proportion of Red balls is p and of Blue balls is 1-p) such that, given these beliefs, the choices of the DM can be rationalized by expected utility? If so, provide the subjective probabilities that rationalize the choices. If not, argue that there are no such probabilities. [Note: Because the outcomes are only $0 and $100, risk preferences play no role here. That is, all utility functions for which u($100) > u($0) are observationally equivalent on these choices. Notice that in order for the curvature of the utility function to be relevant, one would need to consider at least three wealth levels. This is the reason that I am asking only about probabilities and not also about the utility function in this question.]

10.74 What are the eigenfunction, good kuantum numbers, and eigenergy degeneracies for the following configurations?

(a) A free particle moving in three dimensions in (i) Cartesian (ii) spherical and (iii) cylindrical coordinates

(b) A particle confined to a (i) cubica box (ii) spherical box (iii) cylindrical box

       (c) In each case state which type of degeneracy is described (see. Section 8.5) for infinite degeneracies discuss the nature of these infinities (i.e continuous or discrete)

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2. Explain whether the industry you have chosen has a small or large impact on the entire economy. Why or why not? Should a macroeconomist be concerned about what is happening in your industry? Explain.

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explain how process improvements using control charts are used to support areas of industry research, academic research, and scientific research. explain in detail