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【正文】 add the individual probabilities OR use the Addition Rule for probabilities. P(A?B) = P(A) + P(B) P(A ? B) = 1/2 + 1/2 1/4 = 3/4 1 2 3 4 5 6 1 1/36 1/36 1/36 1/36 1/36 1/36 2 1/36 1/36 1/36 1/36 1/36 1/36 3 1/36 1/36 1/36 1/36 1/36 1/36 4 1/36 1/36 1/36 1/36 1/36 1/36 5 1/36 1/36 1/36 1/36 1/36 1/36 6 1/36 1/36 1/36 1/36 1/36 1/36 Second Die First Die Addition Rule AgainWhat is the Probability of throwing a four or higher on either the first or second die? 12 Conditional Probability Conditional Probability: Probability that one Event will occur Given that Another Event has already occurred. To calculate a Conditional probability – Revise Original Sample Space to Account for New Information Eliminates Certain Outes Result: P(A | B) = P(A and B) P(B) 13 1 2 3 4 5 6 1 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 2 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 3 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 4 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 1 /6 5 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 6 1 /36 1 /36 1 /36 1 /36 1 /36 1 /36 Second Die First Die Conditional Probability What is the Probability of throwing at least an 8 if the first die is a 4? Answer: Note we know we are in the row associated with a 4 for the first die. There are only 3 ways of the possible six outes for the second die that yield a sum ? 8. Hence P(A|B) = = P(A?B)/P(B) = (3/36)/(6/36) =.5 14 1. Event Occurrence Does Not Affect Probability of Another Event Toss first die, what does its oute tell you about the likely results in tossing the second die? Answer: Nothing. The two die are independent 2. Causality Not Implied 3. Tests For Statistical Independence P(A | B) = P(A) P(A and B) = P(A)*P(B) Statistical Independence 15 Multiplication Rule 1. Used to Get Compound Probabilities for Intersection of Events Called Joint Events 2. P(A and B) = P(A ? B) = P(A)*P(B|A) = P(B)*P(A|B) 3. For Independent Events: P(A and B) = P(A ? B) = P(A)*P(B) 16 Answer: The result on the first die is independent of the second die. We use The multiplication rule to find prob. of A and B Hence P(A ? B) = = P(A)*P(B) = (1/6)*(1/6) = (1/36) 1 2 3 4 5 6 1 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 2 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 3 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 4 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 5 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 1/3 6 Second Die First Die Multiplication Rule with Independence What is the Probability of throwing “snake eyes” . a 1 on each die? 17 Discrete .’s Summary Measures Expected Value Mean of Probability Distribution Weighted Average of All Possible Values ?X = E(X) = ?Xi P(Xi) = p(X1) X1+ p(X2) X2+ … + p(X n) Xn Variance Weighted Average Squared Deviation about Mean ?X2 = E[ (Xi?X?????????(Xi?X???P(Xi) = p(X1)(X1 ?X)2 + p(X2 )(X2 ?X)2 + … + p(X n) )(Xn ?X)2 Standard deviation Square root of the variance. Measure of risk shows dispersion of possible outes around expected level of outes. 18 Mean/Variance Calculation Table X i P(X i ) X i P(X i ) X i ?? (X i ??) 2 ( X i ??) 2 P( X i ) Total ??X i P(X i ) ??( X i ??) 2 P( X i ) 19 Asset Expected Return amp。 Risk Calculate the Expected Returns of alternative assets. Rki= Return to Asset k if Event i occurs P(Rki) = Probability Event i occurs. E conomi c C ond i t i ons P robab i l i t y A ss et A %
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