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

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Presentation transcript:

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

Consider the following group of people in a room: AgeNumber

Histogram Form

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

Consider the following group of people in a room: Total people = 14 AgeNumberProbability ?

Consider the following group of people in a room: Total people = 14 AgeNumberProbability /

Consider the following group of people in a room: Total people = 14 AgeNumberProbability / ?

Consider the following group of people in a room: Total people = 14 AgeNumberProbability / /14

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

Consider the following group of people in a room: Total people = 14 AgeNumberProbability 1411/ / / / / /14

Probability Histogram

Number Histogram

NB: The probabilities for ages not listed are all zero Total people = 14 AgeNumberProbability 1411/ / / / / /14

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

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

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

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

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

Consider a different room with different people: AgeNumber Total people = 15

Consider a different room with different people: AgeNumberProbability 1933/ / / / / /15 Total people = 15

Red Room Numbers

Red Room Probabilities

Combine Red and Blue rooms Total people = 29 AgeNumberProbability 1411/ / / / / / / / /29

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.

Expectation Values

Most probable answer = 25 Median = 23 Average = 21

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.

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

Average (number squared) AgeNumber(Number) 2 Probability 14111/ / / / / /14

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

The “width” of a probability distribution

Discuss eqns 1.10 through 1.13 at board

Continuous Variables

Why not measure age in weeks?

Blue room in weeks

Conclusion: Blue room in weeks not very useful/intuitive

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).