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Working with ProbabilitiesPhysics 115a (Slideshow 1)
A. Albrecht
These slides related to Griffiths section 1.3
Consider the following group of people in a room:
Age Number
14 1
15 1
16 3
22 2
24 2
25 5
Total people = 14
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1
15 1 ?
16 3
22 2
24 2
25 5
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1
15 1 1/14
16 3
22 2
24 2
25 5
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1
15 1 1/14
16 3
22 2
24 2
25 5 ?
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1
15 1 1/14
16 3
22 2
24 2
25 5 5/14
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1 ?
15 1 1/14
16 3 ?
22 2 ?
24 2 ?
25 5 5/14
Consider the following group of people in a room:
Total people = 14
Age Number Probability
14 1 1/14
15 1 1/14
16 3 3/14
22 2 2/14
24 2 2/14
25 5 5/14
NB: The probabilities for ages not listed are all zero
Total people = 14
Age Number Probability
14 1 1/14
15 1 1/14
16 3 3/14
22 2 2/14
24 2 2/14
25 5 5/14
Assuming Age<20, what is the probability of finding each age?
Total people = 14
Age Number Probability
14 1 ?
15 1 ?
16 3 ?
22 2 ?
24 2 ?
25 5 ?
Assuming Age<20, what is the probability of finding each age?
Total people = 14
Age Number Probability
14 1 ?
15 1 ?
16 3 ?
22 2 0
24 2 0
25 5 0
Assuming Age<20, what is the probability of finding each age?
Total people = 14
Age Number Probability
14 1 1/5
15 1 1/5
16 3 3/5
22 2 0
24 2 0
25 5 0
Total people = 14
Age Number Probability
14 1 1/14
15 1 1/14
16 3 3/14
22 2 2/14
24 2 2/14
25 5 5/14
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
Age Number Probability
14 1 1/5
15 1 1/5
16 3 3/5
22 2 0
24 2 0
25 5 0
Related to collapse of the waveunction (“changing the question”)
Consider a different room with different people:
Age Number
19 3
20 2
21 5
22 3
24 1
25 1
Total people = 15
Consider a different room with different people:
Age Number Probability
19 3 3/15
20 2 2/15
21 5 5/15
22 3 3/15
24 1 1/15
25 1 1/15
Total people = 15
Combine Red and Blue rooms
Total people = 29
Age Number Probability
14 1 1/29
15 1 1/29
16 3 3/29
19 3 3/29
20 2 2/29
21 5 5/29
22 2+3 5/29
24 2+1 3/29
25 5+1 6/29
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.
0 10 20 30 400
0.1
0.2
0.3
0.4
Age
Pro
ba
bili
ty
Most probable answer = 25Median = 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):
0
0
j
jtot
jN j
j jP jN
• Standard QM notation
• Called “expectation value”
• NB in general (including the above) the “expectation value” need not even be possible outcome.
Average (number squared)
Age Number (Number)2 Probability
14 1 1 1/14
15 1 1 1/14
16 3 9 3/14
22 2 4 2/14
24 2 4 2/14
25 5 25 5/14
2
02 2
0
449.4j
jtot
j N j
j j P jN
In general, the average (or expectation value) of some function f(j) is
0j
f j f j P j
Careful: In general 2 2j j
441 449.4
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).