QABE_Notes09 - QABE Lecture 9 Probability II Probability in...

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QABE Lecture 9 Probability II: Probability in action School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Probability Trees 2 3 Rules of Probability 2 3.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 Probability Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.4 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Bayes’ Formula 5 1 Introduction Continuing our two-part look at probability, we now get our teeth into more complicated scenarios of probability. In particular, we will be interested to consider what ‘makes sense’ when we think about how two, or three, or more events occur in time. Along the way, we’ll notice that the ordering of the events matters sometimes, but not always; it’s a question of dependence . Most of our time will be spent drawing and understanding a very useful visualisation of probability problems known as probability trees . It should be pointed out that there is more than one way of thinking about probability, for example, seeing probability in terms of sets is another helpful one. However, in the interests of time, and because trees are especially useful for introducing more complicated topics like Bayes’ Formula , we’ll stick to the trees! Agenda 1. Probability by the trees; 2. Rules of probability for independent events; 3. Conditional probability and Bayes’ Formula. 1
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ECON 1202/ECON 2291: QABE c School of Economics, UNSW 2 Probability Trees Re-introducing Trees We looked at Probability Trees last time, as a diagram with levels. A B C A 1 2 3 1 2 A A B C B C 3 B C Consider drawing cards from a pack, two mutually exclusive events associated with drawing a card are, Black Pr( Black ) Red Pr( Red ) By the cards... In fact, any number of mutually exclusive events can be represented with a tree; Consider drawing cards, but this time, focussing on the suit, Pr( ) 13 52 Pr( ) 13 52 Pr( ) 13 52 Pr( ) 13 52 With a 52 card pack, there are 13 cards of each suit, giving a probability of each event occuring (drawing a card of that suit) equal to 13 52 .
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