Probability Applications
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# Probability Applications ## 10.1 Probabilistic Method Show existence of an object by proving the probability of its existence is positive. **Example:** Lower bound for Ramsey numbers: If $\binom{n}{k} 2^{1-\binom{k}{2}} n$.
Probability Applications
10.1 Probabilistic Method
Show existence of an object by proving the probability of its existence is positive.
Example: Lower bound for Ramsey numbers: If , then .
10.2 Expectation and Variance
Linearity of expectation: (always, even for dependent variables).
Example: Expected number of fixed points in a random permutation of elements: Let if is fixed, 0 otherwise. . By linearity, .
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