STA6166 F05-7 Introduction to Probability

STA6166 F05-7 Introduction to Probability - Topic (7)...

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Topic (7) – INTRODUCTION TO PROBABILITY 7-1 Topic (7) – Introduction to Probability EXAMPLE You buy a lottery ticket. There are two possible outcomes: you’ll either win money or you won’t. How likely is it you’ll win? Defn: Let E denote an outcome to an experiment. The PROBABILITY that the event E occurs is the likelihood or chance of observing that particular outcome. We denote the probability of an event as P(E) or Pr(E) or Prob(E) EXAMPLE Suppose you’re told the chances of winning the lottery are one in 7.2 million. Hence Pr(winning the lottery with a single ticket) = 1 7 200 000 ,, EXAMPLE The probability that a fair die will show a five when tossed is Pr(Five) = 1 6 or 0.1667 or 16.7%.
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Topic (7) – INTRODUCTION TO PROBABILITY 7-2 Defn: RELATIVE FREQUENCY INTERPRE- TATION: The probability of an event E equals the relative frequency of occurrences of E in an indefinitely long number of identical repetitions of the experiment. Pr(E) = # times E occurs . Total # of repetitions EXAMPLE How do I know that Pr(five) = 0.167 for a 6-sided die? Well if I really want to know this for a die in my hand, I should roll the die 1000s of times and record how many times a five appears. EXAMPLE We have a population whose frequency distribution follows the empirical rule: Y 4.2 5 3 .7 3.2 2 .2 1.7 1 . 25 -. - 7 -1 -2 -3 Histogram Frequency 200 100 0 Std. Dev = 1.00 Mean = .04 N = 1500.00
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Topic (7) – INTRODUCTION TO PROBABILITY 7-3 So, according to the empirical rule, ~ 68% of the population have Y-values between 0.04 - 1.00 = - 0.96 and 0.04 + 1.00 = 1.04 ~ 95% of the population have Y-values between 0.04 - 2.00 = - 1.96 and 0.04 + 2.00 = 2.04 > 99% of the population have Y-values between 0.04 - 3.00 = - 2.96 and 0.04 + 3.00 = 3.04 What is the probability that a single selection of a unit from this population would result in an observed value greater that 2.04?
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STA6166 F05-7 Introduction to Probability - Topic (7)...

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