Transcription of 3. Conditional probability & independence Conditional ...
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3. Conditional probability & independenceConditional probabilities Question: How should we modifyP(E)if welearn that eventFhas occurred? Derivation: Suppose we repeat the experimentntimes. Letn(E F)be the number of timesthat bothEandFoccur, andn(F)the numberof timesFoccurs. The proportion of timesEoccurs only countingtrials whereFoccurs isn(E F)n(F)=n(E F)/nn(F)/n P(E F)P(F). Definition: the Conditional probability ofEgivenFisP(E|F) =P(E F)P(F), forP(F)>01 Example 1. 27 students out of a class of 43 areengineers. 20 of the students are female, of whom7 are engineers. Find the probability that arandomly selected student is an engineer giventhat she is 2. Deal a 5 card poker hand, and letE={at least 2 aces},F={at least 1 ace},G={hand contains ace of spades}.(a) FindP(E)(b) FindP(E|F)(c) FindP(E|G)3 The Multiplication Rule Re-arranging the Conditional probabilityformula givesP(E F) =P(F)P(E|F)This is often useful in computing the probabilityof the intersection of Draw 2 balls at random withoutreplacement from an urn with 8 red balls and 4white balls.
3. Conditional probability & independence Conditional Probabilities • Question: How should we modify P(E) if we learn that event F has occurred? • Derivation: Suppose we repeat the experiment n times. Let n(E ∩ F) be the number of times that both E and F occur, and n(F) the number of times F occurs. • The proportion of times E occurs only counting trials where F occurs is
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