# 03 - Probability Introduction Reminder Cryptography and...

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Introduction Probability Reminder Cryptography and Protocols Andrei Bulatov

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Cryptography and Protocols - Probability 3-2 Sample Space and Outcomes Experiment s and outcomes Sample space is the set of all possible outcomes Examples - flipping a coin = {heads, tails} - flipping a pair of coins = {HH, HT, TH, TT} - horse race (7 horses) = {all 7! permutations of (1,2,3,4,5,6,7)} - tossing two dice = {11,12, … , 66} flipping k coins = {0,1} k
Cryptography and Protocols - Probability 3-3 Events Event is any subset of the sample space Examples - any outcome is an event (a 1-element subset) - getting even number of heads when flipping a pair of coins - horse no. 4 came second - getting at least one 3 when tossing two dice Algebra of events - union of events - intersection - complement - mutually exclusive events

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Cryptography and Protocols - Probability 3-4 Probability: Case of Equally Likely Outcomes If all the outcomes are equally likely, then the probability of event A equals Pr[A] = m/n where m is the number of outcomes in A, and n the total number of outcomes Examples - getting even number of heads when flipping a pair of coins - horse no. 4 came second - getting at least one 3 when tossing two dice
Cryptography and Protocols - Probability 3-5 Probability: General Case The probability of event A is a positive number Pr[A] Axioms: - 0 ≤ Pr[A] ≤ 1 - Pr[ ] = 1 - for any events A, B such that AB = Pr[A B] = Pr[A] + Pr[B] Examples - what is the probability to get both heads and tails flipping 3 identical coins?

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03 - Probability Introduction Reminder Cryptography and...

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