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: AgeNumber 141 151 163 222 242 255

3. Histogram Form

4. Consider the following group of people in a room: AgeNumber 141 151 163 222 242 255 Total people = 14

5. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 141 151? 163 222 242 255

6. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 141 1511/14 163 222 242 255

7. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 141 1511/14 163 222 242 255?

8. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 141 1511/14 163 222 242 2555/14

9. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 141? 1511/14 163? 222? 242? 2555/14

10. Consider the following group of people in a room: Total people = 14 AgeNumberProbability 1411/14 1511/14 1633/14 2222/14 2422/14 2555/14

11. Probability Histogram

12. Number Histogram

13. NB: The probabilities for ages not listed are all zero Total people = 14 AgeNumberProbability 1411/14 1511/14 1633/14 2222/14 2422/14 2555/14

14. Assuming Age<20, what is the probability of finding each age? Total people = 14 AgeNumberProbability 141? 151? 163? 222? 242? 255?

15. Assuming Age<20, what is the probability of finding each age? Total people = 14 AgeNumberProbability 141? 151? 163? 2220 2420 2550

16. Assuming Age<20, what is the probability of finding each age? Total people = 14 AgeNumberProbability 1411/5 1511/5 1633/5 2220 2420 2550

17. Total people = 14 AgeNumberProbability 1411/14 1511/14 1633/14 2222/14 2422/14 2555/14 Assuming no age constraint, what is the probability of finding each age? Related to collapse of the waveunction (“changing the question”)

18. Assuming Age<20, what is the probability of finding each age? Total people = 14 AgeNumberProbability 1411/5 1511/5 1633/5 2220 2420 2550 Related to collapse of the waveunction (“changing the question”)

19. Consider a different room with different people: AgeNumber 193 202 215 223 241 251 Total people = 15

20. Consider a different room with different people: AgeNumberProbability 1933/15 2022/15 2155/15 2233/15 2411/15 2511/15 Total people = 15

21. Red Room Numbers

22. Red Room Probabilities

23. Combine Red and Blue rooms Total people = 29 AgeNumberProbability 1411/29 1511/29 1633/29 1933/29 2022/29 2155/29 222+35/29 242+13/29 255+16/29

28. 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.

29. Expectation Values

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

31. 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.

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

33. Average (number squared) AgeNumber(Number) 2 Probability 14111/14 15111/14 16393/14 22242/14 24242/14 255 5/14

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

35. The “width” of a probability distribution

39. Discuss eqns 1.10 through 1.13 at board

40. Continuous Variables

41. Why not measure age in weeks?

42. Blue room in weeks

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

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