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Expected Value Advanced is authored in the NetLogo modeling-and-simulation environment. The model is part of ProbLab, a curricular unit designed to enrich student understanding of the domain. The online unit package will include a suite of models, student worksheets, and a teacher guide. Below is an applet of 'Expected Value Advanced.' You can interact with this model by pressing the buttons and changing the slider values to run this model under different settings. Note that this model is still under development and is yet to undergo our rigorous checkout procedure.
Expected Value Advanced studies expected-value analysis under the special condition that the sample size varies. This model extends the ProbLab model Expected Value, where the sample size is fixed.
The analogy utilized for the model is one of a lake with fish swimming around, each type of fish worth a certain number of dollars (1, 2, 3, 4 or 5) [other currencies or point systems apply just as well]. The distribution of types of fish by amount-of-worth -- how many $1 fish, $2 fish, ..., or $5 fish there are in the pond -- is set by the sliders on the left. The more fish there are of a particular kind, say the more $2 fish there are, the higher the chance of catching a $2 fish in a random sample. With the sliders, we can set the distribution of fish by type and, therefore, the chance of catching each type of fish. That is, the higher you have set a given slider as compared to other sliders (see the % IN POPULATION monitors), the higher the chance of catching a fish with that worth. Note that the more valuable the fish, the lighter its body color. You can press CLICK SELECTION and then click on the screen to "catch" a random sample, or press RANDOM SELECTION and have the computer program do the choosing for you. The computer selects randomly. You, too, can select blindly, if you turn the BLIND? switch to 'On.'
The idea of 'expected value' is that we can formulate an educated guess of the dollar worth of the fish we catch. It's similar to asking, "How much does the average fish cost?" We need to somehow take into account both the chance of getting each type of fish and its dollar value. The computer program will do much of the calculations for us, but here's the gist of what it does:
Let's say that the ratio units we set up for $1, $2, $3, $4 and $5 were, respectively, 1 : 6 : 5 : 0 : 4. The number '5,' for example, indicates our ratio setting for fish worth 3 dollars. You can immediately see that the chance of getting a $2-fish is a greater than the chance of getting a $3-fish, because the chance of getting a $2-fish has more ratio units than the $3-fish. But in order to determine just how big the chance is of getting each type of fish, we need to state the ratio units relative to each other. We need a common denominator. In this particular setting, there is a total of 16 'ratio units': 1 + 6 + 5 + 0 + 4 = 16. Now we can say that there is a 4/16 chance of getting a $5-fish. That is a 25% chance of catching a fish that is worth exactly 5 dollars. We can also say that this relative proportion of $5-fish in the lake contributes .25 * 5, that is, $1.25, to the mean value of a single fish in the lake. Similarly, we can say there is a 5-in-16 chance of getting a $3-fish, a 4-in-16 chance of getting a $5-fish, etc. If we sum up all products of 'value' and 'probability,' we get the expected value per single fish:
(1 * 1/16) + (2 * 6/16) + (3 * 5/16) + (4 * 0/16) + (5 * 4/16) = 48/16 = 3 dollars per fish.
This tells us that if you pick any single fish, you should expect, on average, to get a value of 3 dollars. If you were to select a sample of 4 fish, then you would expect, on average, to pocket (4 fish * 3 dollars-per-fish =) 12 dollars. Note, though, that the sampling in this model is of arrays, e.g., a 2-by-2 array of 4 squares. There are as many fish in this model as there are squares. One might expect to catch 4 fish when one samples from 4 squares. However, when the WANDER button is pressed, the fish wander randomly, and so sometimes 4 squares have more than 4 fish and sometimes they have less. You can think of each selection as a fishers' net that is dipped into the lake -- the fisher doesn't know how many fish will be in the net. This feature of the model creates variation in sample size. Thus, one idea that this model explores is that even under variation in sample size, we still receive outcomes that correspond to the expected value that we calculate before taking samples.
Questions to Ponder
The WANDER button determines whether the fish can wander in the lake as you sample from them. Why is it that the graphs of Number per Sample and Value per Sample each approximate a normal curve quicker when the WANDER button is set to 'On' as compared to when it is set to 'Off?'
[last updated January 4, 2005]