1Chapter 5: ProbabilityChapter 6: ProbabilityDistributionsRead Chapter 6ProbabilityThe probability of an event is defined as its relative frequency.Relative Frequency DefinitionThat is, for an event A,P(A)= number of ways A happenstotal number of outcomesEmpirical/mechanistic approach to probabilityAmerican 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, theexpected 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 valueof a one dollar bet is $0.9474. On an European Roulette table: −$1(36/37) + $35(1/37) = −$0.0270. The expected valueof a one dollar bet is now $0.9730.Properties of ProbabilityFor every event A in Ω(the set of all events),0 ≤P(A) ≤1P(Ω)=1The set of outcomes that are not in event A is called the complement of A, Ac.P(A) = 1 – P(Ac)Properties of ProbabilityFor two events A and BP(AUB) = P(A) + P(B) – P(A∩B).http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Venn.htmlExample: 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 52525226
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