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Probability Review. Part I. Design and Analysis of Algorithms I. Topics Covered. Sample spaces Events Random variables Expectation Linearity of Expectation See also: Lehman-Leighton notes (free PDF) Wikibook on Discrete Probability. Concept #1 – Sample Spaces. Concept #2 – Events. - PowerPoint PPT Presentation
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Probability Review
Part IDesign 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
Tim Roughgarden
Concept #1 – Sample Spaces
Tim Roughgarden
Concept #2 – Events
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?
1𝑛141234
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?
Tim Roughgarden
Concept #2 – Events
Tim Roughgarden
Concept #3 - Random Variables
Tim Roughgarden
Concept #4 - Expectation
6 .577.58
What is the expectation of the sum of two dice?
𝑛4𝑛3𝑛2
3𝑛4
Which of the following is closest to the expectation of the size of the subarray passed to the first recursive call in QuickSort?
Tim Roughgarden
Concept #4 - Expectation
Tim Roughgarden
Concept #5 – Linearity of Expectation
Tim Roughgarden
Linearity of Expectation (Proof)
Tim Roughgarden
Example: Load Balancing
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Load Balancing Solution
Tim Roughgarden
Load Balancing Solution (con’d)