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

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Concept #1 – Sample Spaces

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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?

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?

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Concept #2 – Events

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Concept #3 - Random Variables

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Concept #4 - Expectation

6 .577.5

8

What is the expectation of the sum of two dice?

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?

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Concept #4 - Expectation

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Concept #5 – Linearity of Expectation

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Linearity of Expectation (Proof)

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Example: Load Balancing

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Load Balancing Solution

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Load Balancing Solution (con’d)