Chapter5_2 - Probability Chapter 5: Probability Chapter 6:...

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1 Chapter 5: Probability Chapter 6: Probability Distributions Read Chapter 6 Probability The probability of an event is defined as its relative frequency. ± Relative Frequency Definition That is, for an event A, P(A)= number of ways A happens total number of outcomes ± Empirical/mechanistic approach to probability American or European Roulette? A common application of expected value is in gambling. For example, an American Roulette wheel has 38 equally likely outcomes. A winning bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). So considering all 38 possible outcomes, the expected profit resulting from a $1 bet on a single number is: $1(37/38) + $35(1/38) = $0.0526. Therefore one expects, on average, to lose over five cents for every dollar bet, and the expected value of a one dollar bet is $0.9474. On an European Roulette table: $1(36/37) + $35(1/37) = $0.0270. The expected value of a one dollar bet is now $0.9730. Properties of Probability ² For every event A in (the set of all events), ± 0 P(A) 1 ± P( )=1 ² The set of outcomes that are not in event A is called the complement of A, A c . ± P(A) = 1 – P(A c ) Properties of Probability For two events A and B P(AUB) = P(A) + P(B) – P(A B). http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Venn.html ² Example: If a single card is randomly drawn from an ordinary deck of 52 cards, find the probability that it will be a club or a face card. P(E F) = P(E) + P(F) – P(E F) P(club face) = P(club) + P(face) – P(club face) P(club face) = 13 + 12 -3 = 22 = 11 . 52 52 52 52 26
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2 Conditional Probability ± For events A and B, the conditional probability of event A given that event B has occurred, is: ± P(A|B) is read as “the probability of event A given event B.” The vertical slash represents the word “given”. Of the times that B occurs, P(A|B) is the proportion of times that A also occurs ) ( ) ( ) | P( B P B and A P B A = Conditional Probability Since P(A|B) = P(A B) P(B) then if follows that: P(A B) = P(A|B) P(B) (multiplication rule) If A1 occurs before A2 then we know: P(A1 and A2) = P(A2|A1) P(A1) More generally: P(A1 and A2 and A3) = P(A3|A2 and A1) P(A2|A1) P(A1) Is a “Coincidence” Truly an Unusual Event? ± The law of very large numbers states that if
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This note was uploaded on 01/09/2010 for the course ILRST 2100 at Cornell.

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Chapter5_2 - Probability Chapter 5: Probability Chapter 6:...

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