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# handout1 - ISyE8843A Brani Vidakovic Handout 1 1...

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ISyE8843A, Brani Vidakovic Handout 1 1 Probability, Conditional Probability and Bayes Formula The intuition of chance and probability develops at very early ages. 1 However, a formal, precise definition of the probability is elusive. If the experiment can be repeated potentially infinitely many times, then the probability of an event can be defined through relative frequencies. For instance, if we rolled a die repeatedly, we could construct a frequency distribution table showing how many times each face came up. These frequencies ( n i ) can be expressed as proportions or relative frequencies by dividing them by the total number of tosses n : f i = n i /n. If we saw six dots showing on 107 out of 600 tosses, that face’s proportion or relative frequency is f 6 = 107 / 600 = 0 . 178 As more tosses are made, we “expect” the proportion of sixes to stabilize around 1 6 . Famous Coin Tosses: Buffon tossed a coin 4040 times. Heads appeared 2048 times. K. Pearson tossed a coin 12000 times and 24000 times. The heads appeared 6019 times and 12012, respectively. For these three tosses the relative frequencies of heads are 0.5049, 0.5016,and 0.5005. What if the experiments can not be repeated? For example what is probability that Squiki the guinea pig survives its first treatment by a particular drug. Or “the experiment” of you taking ISyE8843 course in Fall 2004. It is legitimate to ask for the probability of getting a grade of an A . In such cases we can define probability subjectively as a measure of strength of belief. Figure 1: A gem proof condition 1913 Liberty Head nickel, one of only five known and the finest of the five. Collector Jay Parrino of Kansas City bought the elusive nickel for a record \$1,485,000, the first and only time an American coin has sold for over \$1 million. Tutubalin’s Problem. In a desk drawer in the house of Mr Jay Parrino of Kansas City there is a coin, 1913 Liberty Head nickel. What is the probability that the coin is heads up? The symmetry properties of the experiment lead to the classical definition of probability. An ideal die is symmetric. All sides are “equiprobable”. The probability of 6, in our example is a ratio of the number of favorable outcomes (in our example only one favorable outcome, namely, 6 itself) and the number of all possible outcomes, 1/6. 2 1 Piaget, J. and Inhelder B. The Origin of the Idea of Chance in Children , W. W. Norton & Comp., N.Y. 2 This definition is attacked by philosophers because of the fallacy called circulus vitiosus . One defines the notion of probability supposing that outcomes are equi probable . 1

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( Frequentist ) An event’s probability is the proportion of times that we expect the event to occur, if the experiment were repeated a large number of times. ( Subjectivist ) A subjective probability is an individual’s degree of belief in the occurrence of an event.
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