probability review part i design and analysis of algorithms i

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Probability Review Part I Design and Analysis of Algorithms I

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Page 1: Probability Review Part I Design and Analysis of Algorithms I

Probability Review

Part IDesign and Analysis

of Algorithms I

Page 2: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Topics Covered• Sample spaces• Events• Random variables• Expectation• Linearity of ExpectationSee also:• Lehman-Leighton notes (free PDF)• Wikibook on Discrete Probability

Page 3: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #1 – Sample Spaces

Page 4: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #2 – Events

Page 5: Probability Review Part I Design and Analysis of Algorithms I

1361121612

Consider the event (i.e., the subset of outcomes for which) “the sum of the two dice is 7”. What is the probability of this event?

Page 6: Probability Review Part I Design and Analysis of Algorithms I

1𝑛

141234

Consider the event (i.e., the subset of outcomes for which) “the chosen pivot gives a 25-75 split of better”. What is the probability of this event?

Page 7: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #2 – Events

Page 8: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #3 - Random Variables

Page 9: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #4 - Expectation

Page 10: Probability Review Part I Design and Analysis of Algorithms I

6 .577.5

8

What is the expectation of the sum of two dice?

Page 11: Probability Review Part I Design and Analysis of Algorithms I

𝑛4

𝑛3

𝑛2

3𝑛4

Which of the following is closest to the expectation of the size of the subarray passed to the first recursive call in QuickSort?

Page 12: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #4 - Expectation

Page 13: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Concept #5 – Linearity of Expectation

Page 14: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Linearity of Expectation (Proof)

Page 15: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Example: Load Balancing

Page 16: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Load Balancing Solution

Page 17: Probability Review Part I Design and Analysis of Algorithms I

Tim Roughgarden

Load Balancing Solution (con’d)