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Understanding Probabilities in Physics: Analyzing Age Distribution in Groups

This slideshow by A. Albrecht presents a detailed exploration of probabilities related to age distributions in groups of people. With examples involving rooms of different sizes, it illustrates how to calculate probabilities for various age scenarios, including the impact of constraints like age < 20. Key lessons include normalization, re-normalization, and the importance of expectation values in probability distributions. By analyzing combinations of groups (e.g., red and blue rooms), it emphasizes intuitive applications of probability in physics contexts.

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Understanding Probabilities in Physics: Analyzing Age Distribution in Groups

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  1. Working with Probabilities Physics 115a (Slideshow 1) A. Albrecht These slides related to Griffiths section 1.3

  2. Consider the following group of people in a room:

  3. Histogram Form

  4. Consider the following group of people in a room: Total people = 14

  5. Consider the following group of people in a room: Total people = 14

  6. Consider the following group of people in a room: Total people = 14

  7. Consider the following group of people in a room: Total people = 14

  8. Consider the following group of people in a room: Total people = 14

  9. Consider the following group of people in a room: Total people = 14

  10. Consider the following group of people in a room: Total people = 14

  11. Probability Histogram

  12. Number Histogram

  13. NB: The probabilities for ages not listed are all zero Total people = 14

  14. Assuming Age<20, what is the probability of finding each age? Total people = 14

  15. Assuming Age<20, what is the probability of finding each age? Total people = 14

  16. Assuming Age<20, what is the probability of finding each age? Total people = 14

  17. Assuming no age constraint, what is the probability of finding each age? Related to collapse of the waveunction (“changing the question”) Total people = 14

  18. Assuming Age<20, what is the probability of finding each age? Related to collapse of the waveunction (“changing the question”) Total people = 14

  19. Consider a different room with different people: Total people = 15

  20. Consider a different room with different people: Total people = 15

  21. Red Room Numbers

  22. Red Room Probabilities

  23. Combine Red and Blue rooms Total people = 29

  24. Lessons so far • A simple application of probabilities • Normalization • “Re-Normalization” to answer a different question • Adding two “systems”. • All of the above are straightforward applications of intuition.

  25. Expectation Values

  26. Most probable answer = 25 Median = 23 Average = 21

  27. Most probable answer = 25 Median = 23 Average = 21 Lesson: Lots of different types of questions (some quite similar) with different answers. Details depend on the full probability distribution.

  28. Average (mean): • Standard QM notation • Called “expectation value” • NB in general (including the above) the “expectation value” need not even be possible outcome.

  29. Average (number squared)

  30. Careful: In general In general, the average (or expectation value) of some function f(j) is

  31. The “width” of a probability distribution

  32. Discuss eqns 1.10 through 1.13 at board

  33. Continuous Variables

  34. Continuous Variables Why not measure age in weeks?

  35. Blue room in weeks

  36. Blue room in weeks

  37. Conclusion: Blue room in weeks not very useful/intuitive

  38. Another case where a measure of age in weeks might by useful: The ages of students taking health in the 8th grade in a large school district (3000 students).

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