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Unformatted text preview: STAT 200 Chapter 4 Probability: The Study of Randomness Randomness (Section 4.1) Consider throwing a thumbtack. The outcome can be a ‘pointup’ or ‘pointdown’, but we cannot predict with certainty which outcome will occur before throwing the thumbtack. This is a random phenomenon . What is the chance or probability that the thumbtack lands with ‘pointup’? Let’s throw a thumbtack for a few times and look at the outcomes. Toss # Outcome (D‘pointdown’, U‘pointup’) 1 D 2 D 3 U 4 D 5 U Based on the five attempts or trials , we have 40% ‘pointup’ and 60% ‘pointdown’. Can we conclude the chance of landing with ‘pointup’ is 40%? Suppose for each additional toss, we compute the relative frequency of ‘pointup’ (i.e., the % of landing with ‘pointup’) from all the available tosses. Total # tosses #D’s #U’s relative freq. of U 1 1 0% (0/1) 2 2 0% (0/2) 3 2 1 33% (1/3) 4 3 1 25% (1/4) 5 3 2 40% (2/5) If we continue to throw the thumbtack and keep track of the relative frequency of ‘pointup’, we will see that the relative frequency eventually stabilizes at a single value. 1 We can see it in the plot of the relative frequency against the number of tosses. The relative frequency of ‘pointup’ stays relatively constant at around 60% for a large num ber of tosses. Independent trials: We have done a series of tosses of the thumbtack. Suppose the thumbtack lands with ‘pointup’ in this toss. Does this outcome affect the outcome of the next toss? No. We say the tosses are independent . Trials are said to be independent if the outcome of a trial does not influence the outcome of the other trials. Law of large numbers (LNN) The Law of Large Numbers states that, given that the repeated trials are independent, the long run relative frequency of an outcome approaches the true relative frequency of the outcome as the number of trials increases. The tosses are independent. The long run relative frequency of “pointup” gives the proba bility of tossing a ‘pointup’, which is 60%. 2 Probability concepts (Section 4.2) • A sample space S is the set of of all possible outcomes of a random phenomenon. • An event is an outcome or a combination of outcomes from a random phenomenon. We denote an event by an uppercase letter, e.g., A, B, C. e.g., Tossing a ‘pointup’ is an event. Tossing a ‘pointdown’ is another event. Tossing two ‘pointup”s in two tosses is also an event. • The notation P ( A ) denotes the probability of an event A that will occur. Properties of P ( A ): 1. 0 ≤ P ( A ) ≤ 1 P ( A ) = 0 implies event A is impossible. P ( A ) = 1 implies event A is certain. The larger the P ( A ), the higher the chance that the event A will occur....
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 Spring '10
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 Normal Distribution, Probability, Standard Deviation, Probability theory, probability density function

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