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Probability (Basic (Definition (Sample space(S): a set of all possible…
Probability
Basic
Example
Assumption: All outcomes are equally likely, think of the probability of event A as the number of outcomes in A divided by the number of outcomes in S events
Event A = {1, 2}, P(A) = 2/6
Event B = {6}, P(B) = 1/6
Complement of A = A ^ c = {3,4,5,6}
P(A^c) = 1 - P(A) = 2/3
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Independent Event
Example
Rolling a pair of dice
P(get 6 in first die) = 1/6
P(get 6 in second die) = 1/6
P(get 6 in first die n get 6 in the second die) = 1/6 x 1/6 = 1/36
Definition
Two events A and B are independent if learning that B has occured does not change the probability of A occuring
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P(A | B) = P(A n B) / P(B), so if P(B) != 0
P(A n B) = P(A | B) x P(B)
P(A n B) = P(A) x P(B) if A and B are independent
Conditional
Example
Rolling a die: S = {1,2...6}
Event A = getting the number 6, P(A) = 1/6
Event B = getting an even number, P(B) = 3/6
P(A and B) = P(A n B) = 1/6
P(A | B) = P(A n B) / P(B) = (1/6) / (1/2) = 1/3
Definition
Conditional probability of event A given event B has occured
P(A | B) = P(A and B) / P(B) = P(A n B) / P(B) = P(A n B) / P(B), if P(B) != 0
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