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Probability and Mean Calculations in Quantum Mechanics

Explore calculations involving mean, median, waiting times, probabilities, and quantum mechanics concepts such as atomic structure and electron orbitals.

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Probability and Mean Calculations in Quantum Mechanics

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  1. A spinner from a board game randomly indicates a real number between 0 and 50. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length. Find the mean. • 7.07 • 49 • 25 • 50 • 2,500 • 12.5

  2. Suppose the average waiting time for a customer's call to be answered by a company representative (modeled by exponentially decreasing probability density functions) is 30 minutes. Find the median waiting time. • 20.79 minutes • 24.95 minutes • 18.71 minutes • 16.64 minutes • 31.19 minutes • 22.87 minutes

  3. The manager of a fast-food restaurant determines that the average time that her customers wait for service is 5 minutes. Find the probability that a customer is served within the first 4 minutes. • 0.77 • 0.61 • 0.66 • 0.44 • 0.28 • 0.55

  4. Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. If the target weight is 500 g, what is the probability that the machine produces a box with less than 475 g of cereal? Round your answer to four decimal places. • 0.0186 or 1.86% • 0.0018 or 0.18% • 0.0062 or 0.62% • 0.1056 or 10.56%

  5. The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a "cloud" of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or 1s-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function {image} where {image} is the Bohr radius ( {image} ). The integral {image} gives the probability that the electron will be found within the sphere of radius r meters centered at the nucleus. Find the probability that the electron will be within the sphere of radius {image} centered at the nucleus. • 0.18 • 0.89 • 0.66 • 0.13 • 0.38 • 0.25

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