Chapter 2 - Chapter 2 Conditional Probability Suppose that...

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Chapter 2 Conditional Probability Suppose that we have partial knowledge of the result of a trial of the experiment we know that event A occurred on the trial but we do not know which outcome in A occurred; only that the outcome is some member of event A Did the event B occur? We cannot say for sure: All we know is that the actual outcome x is a member of A, and all we can say is that B occurred if and only if x is a member of B as well , that is, B occurred if x AB, and B did not occur if x AB c Event A is known to have occurred; but we don’t know which outcome in A occurred Event B has occurred if the actual outcome belongs to AB Event B has not occurred (i.e B c has occurred) if the actual outcome belongs to AB c Since we cannot say for sure whether B occurred or not, we would like a probabilistic description of the situation P(B) and P(B c ) should be updated in view of the partial knowledge that the event A occurred on this particular trial of the experiment Special cases: If A B, then x A x B; hence B occurs whenever A does. We should update the probability of B from P(B) to 1 If AB = ø, x A x B; hence B cannot occur when A does. We should update the probability of B from P(B) to 0 More generally, how should we update P(B) in light of the partial knowledge that event A occurred? We need nomenclature and notation to distinguish between the probability that we originally assigned to the event B and the updated probability of the event B P(B), the number originally assigned as the probability of B, is called the unconditional or a priori probability of B The updated probability of B is called the conditional probability of B given A (or the probability of B given A) and is denoted by P(B | A) Assume P(A) > 0 P(B | A), the conditional probability of B given A, is P(B | A) = P(B A) P(A) = P(BA) P(A) Similarly, P(B c | A) = P(B c A) P(A) A is the conditioning event P(B | A) = P(BA) P(A) P(B c | A) = P(B c A) P(A) These definitions also work in the special cases considered above. Example: If A B BA = A, and B c A = ø A B Hence, P(B | A) = 1; P(B c | A) = 0 © 1997 by Dilip V. Sarwate. All rights reserved. No part of this manuscript may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the author.
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Chapter 2 Conditional Probability 17 Example: If BA = ø, then P(B | A) = 0. Also B c A = A P(B c | A) = 1 A B Example: Two fair dice are rolled. What is the (conditional) probability that the sum of the two faces is 6 given that the two dice are showing different faces? Code phrase: given = we are asked to find conditional probability = {(i, j): 1 i 6, i j 6}. B = {(1,5), (2,4), (3,3), (4,2), (5,1)}; P(B) = 5 36 A = {(i, j): (i, j) OE , i j}; P(A) = 30 36 ; BA = {(1,5), (2,4), (4,2), (5,1)}; P(BA) = 4 36 Hence, P(B|A) = P(BA) P(A) = (4/36) (30/36) = 2 15 .
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