chap4 - 9 E JT Chapter 4 ~ Probability Limiting Relative...

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1 JTE9 Chapter 4 ~ Probability Limiting Relative Frequency Relative Frequency Trials 0 0.1 0.2 0.3 0.4 0.5 0.6 0 200 400 600 800 1000 1200
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2 JTE9 Chapter Goals Learn the basic concepts of probability Learn the rules that apply to the probability of both simple and compound events In order to make inferences, we need to study sample results in situations in which the population is known
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3 JTE9 100 Rolls 1000 Rolls Outcome Frequency Outcome Frequency 0 80 0 690 1 19 1 282 2 1 2 28 4.1 ~ The Nature of Probability Example : Consider an experiment in which we roll two six- sided fair dice and record the number of 3s face up. The only possible outcomes are zero 3s, one 3, or two 3s. Here are the results after 100 rolls, and after 1000 rolls:
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4 JTE9 Using a Histogram We can express these results (from the 1000 rolls) in terms of relative frequencies and display the results using a histogram: 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Relative Frequency Three’s Face Up
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5 JTE9 Continuing the Experiment If we continue this experiment for several thousand more rolls: 1. The frequencies will have approximately a 25:10:1 ratio in totals 2. The relative frequencies will settle down Use random number tables Use a computer to randomly generate number values representing the various experimental outcomes Key to either method is to maintain the probabilities Note : We can simulate many probability experiments:
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6 JTE9 4.2 ~ Probability of Events Probability that an Event Will Occur : The relative frequency with which that event can be expected to occur The probability of an event may be obtained in three different ways: Empirically Theoretically Subjectively
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7 JTE9 Experimental or Empirical Probability Question: What happens to the observed relative frequency as n increases? Experimental or Empirical Probability: 1. The observed relative frequency with which an event occurs 2. Prime notation is used to denote empirical probabilities: 3. n (A): number of times the event A has occurred 4. n : number of times the experiment is attempted P n n ( ) ( ) A A
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8 JTE9 Example Example: Consider tossing a fair coin. Define the event H as the occurrence of a head. What is the probability of the event H, P (H)? Notes: This does not mean exactly one head will occur in every two tosses of the coin In the long run, the proportion of times that a head will occur is approximately 1/2 1. In a single toss of the coin, there are two possible outcomes 2. Since the coin is fair, each outcome (side) should have an equally likely chance of occurring 3. Intuitively, P (H) = 1/2 (the expected relative frequency)
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9 JTE9 Long-Run Behavior To illustrate the long-run behavior : 1. Consider an experiment in which we toss the coin several times and record the number of heads 2. A trial is a set of 10 tosses 3. Graph the relative frequency and cumulative relative frequency of occurrence of a head 4. A cumulative graph demonstrates the idea of long-run behavior 5. This cumulative graph suggests a stabilizing, or settling down, effect on the observed cumulative probability 6. This stabilizing effect, or long-term average value, is often referred to as the law of large numbers
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  • Spring '09
  • ASATARBAIR
  • Probability, Probability theory, Tree Diagram, Jt

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