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Simulation Experiments: We're going to lay the groundwork for theoretical probability by using experimental probability

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dyllis
18 diapositives · 311 Views · Jul 11, 2014

The groundwork for theoretical probability through experimental simulations. Simulation serves as an imitation of chance behavior based on a model, helping to estimate result likelihoods. The process involves stating the problem, identifying variables, simulating trials, and analyzing outcomes. For instance, a cereal manufacturer's card distribution can be modeled to predict how many boxes are needed to collect all pictures. By executing simulations, insights into probability scenarios are gained, applying assumptions and past experiences to calculate frequencies and make informed conclusions.

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Simulation Experiments: We're going to lay the groundwork for theoretical probability by using experimental probability
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  1. Simulating Experiments We're going to lay the groundwork for theoretical probability by using experimental probability.  

  2. Simulation: imitation of chance behavior based on a model that accurately reflects the experiment under consideration ·Estimate the likelihood of a result of interest ·Not always possible to run an experiment ·Insight into events to help us understand the situation ·What are the chances that…? ·Calculate the relative frequency ·Probability model – theoretical answer ·Model based on opinion and past experience

  3. Steps to completing a Simulation ·Step 1 State the problem or describe the experiment ·Identify component to be repeated ·Clearly state what the response variable is ·Explain how you will model the outcome ·Step 2 State the assumptions ·Ex. Head or tail is equally likely ·Tosses are independent ·Step 3 Assign digits to represent outcomes ·Explain how you assign digits ·Explain how you will simulate the trial ·Step 4 Simulate many repetitions ·Run several trials ·Show work ·Chart ·Table ·List ·Step 5 State your conclusions ·Analyze the response variable ·State your conclusion in the context of the problem

  4. Simulation Suppose a cereal manufacturer put pictures of famous athletes on cards in boxes of cereal in the hope of boosting sales. The manufacturer announces that 20% of the boxes contain a picture of Shaun White, 30% a picture of Lance Armstrong, and the rest a picture of Venus Williams. You want all three pictures. How many boxes of cereal do you expect to have to buy in order to get the complete set?

  5. Step 1:State the Problem/Describe the Experiment ·A single component = selection of a cereal box ·Response variable – How many boxes does it take to get all 3 cards? ·We will pretend to open cereal boxed until we have one of each picture. We do this by looking at each random number and indicating what outcome it represents. We continue until we have all 3 pictures. We will use a calculator to randomly select a cereal box.

  6. Step 2: State the assumptions ·Manufacturers percentages are correct ·Cards are randomly distributed throughout the boxes

  7. Step 3: Assign digits to represent outcomes Use the digits 0 – 9 to represent the cards ·20% Shaun – digits 0,1 ·30% Lance – digits 2,3,4 ·50%Venus – digits 5,6,7,8,9

  8. Step 4: Simulate many repetitions Example chart…

  9. Step 5: State your conclusions ·Based on this simulation, we estimate that customers that want the complete set of sports star pictures will buy an average of 7.8 boxes.

  10. ·5 simulations is inadequate ·20 is a reasonable minimum by hand ·Best to do a few hundred trials

  11. The issue of Two digits vs. One Example: Consider how we'd set up the following two situations. a)Choose a person at random from a group in which 70% are employed. b)Choose a person at random from a group in which 73% are employed Solution: In part (a), we could use one digit from a table of random digits; let 0-6 = employed; let 7-9 be unemployed. In part (b), to be precise, we'd need two digits; let 00-72 be employed and 73-99 be unemployed.

  12. Your turn: Choose one person at random from a group in which 60% are employed, 10% are unemployed, and 30% are not in the labor force. A possible solution: 0-5 employed; 6 - unemployed, 7-9 not in labor force.

  13. Example (5.62): (Calculator) Avioza makes 70% of his free throws in a long season. In a tournament game, he shoots 5 free throws late in the game and misses three. The fans boo, the cheerleaders look away, the coach screams, and there may be some popcorn making its way on to the court very soon. But wait, Avioza cries, I didn't choke. Let's look at this from a statistical vantage point. OK. Let's. Simulate 50 repetitions of 5 shots. Then calculate the relative frequency that Avi will miss 3 or more of his 5 shots. What conclusions can we draw?

  14. 0-6 = A single made shot 7-9 = A single missed shot X= number of made shots out of 5 Look at 5 random digits – count the number of made shots Repeat more than 20 times Record the frequency of the number of made shots per 5 taken Example 9,6,7,4,6 1,2,1,4,9 ETC. (3 made) (4 made) Based on this simulation, the probability of Avi missing 3 or more shots is 11/50 = .22

  15.  END

  16. Homework ·5.60, 5.61, 5.67