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Geometric_Probabilitygeometric_probability

By Larry Wells,2014-12-10 18:43
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Task 1: Measuring Areas of Irregular Regions using Geometric Probability: Hands-on Dice Activity Directions for plotting random points: 1. Throw two dice. 2. The number on the first die determines the column for your square. 3. The number on the second die determines the row of your target square. 4. Put your dot somewhere in the square identified by the column ..

    Task 1: Measuring Areas of Irregular Regions using Geometric

    Probability: Hands-on Dice Activity

    Directions for plotting random points:

    1. Throw two dice.

    2. The number on the first die determines the column for your square.

    3. The number on the second die determines the row of your target square.

    4. Put your dot somewhere in the square identified by the column and row numbers.

    5. Repeat steps 1-4 twenty times, placing a dot in a square for each throw (thus a square

    may end up with several dots in it).

    6. Count the total number of dots on your grid (should equal the total number of throws).

    7. Count the total number of dots inside the J. Record both counts.

    8. Find the ratio of “dots inside the J to total number of dots.” Record this ratio.

    9. Compare this ratio to the ratio of “number of orange squares to total number of squares.”

    10. Continue the experiment for several more rounds and then repeat steps 6-9.

     6

     S 5 e

    c

    o n 4 d

D

     i 3 e

     2

     1

     1 2 3 4 5 6

    F i r s t D i e

Questions to Ponder:

    Think of two different ways to calculate the area of the J as a fraction of the area of the 6X6 grid.

    If you tossed the pair of dice twenty times, how many times would you expect to plot a point in the J? Would you be surprised if only one point landed in the J? Would you be surprised if 6 points landed in the J?

    If you tossed the pair of dice 100 times, how many times would you expect to plot a point in the J? Would you be surprised if only 5 points landed in the J? Would you be surprised if 30 points landed in the J?

    Which of the above results would be most surprising? Why?

    In the figure below would you expect different or similar results for landing a point in an orange square? Explain your answer.

     6

     5

     4

     3

     2

     1

    1 2 3 4 5 6

What about the following arrangement of 6 orange squares?

     6

     5

     4

     3

     2

     1

    1 2 3 4 5 6

    During the session we shall collect all of your results and plot them using two different methods:

    1. A scatterplot showing total number of throws versus number

    in the J.

    2. A dot plot showing the frequency of each ratio obtained by

    each set of 20 throws.

    ; What would you anticipate the scatterplot to look like? ; What would you anticipate the slope of the line of best

    fit to be on this scatterplot? Give a rationale for your

    answer.

    ; How would you expect the ratios to be distributed on the

    dot plot?

    ; What would you expect the median to be of this frequency

    distribution?

    Extensions into Geometry, Algebra and Calculus using the

    TMTMFathom, Dynamic Data software tool.

    These extensions are based on activity 7.4 "Geometry by Probability -- Monte Carlo Methods" from Exploring Algebra 1

    with Fathom, published by Key Curriculum Press. The following directions assume familiarity with the Fathom software.

    A. Estimating the area of a Circle and finding an estimate for the ratio, Pi.

    1. The area of a quarter circle can be estimated by plotting

    random points in a square of side-length equal to the

    radius of the quarter circle, and then determining which

    points lie inside the quarter circle inscribed inside the

    square and taking the ratio of these points to the total

    number of points (just as we did with the dice toss to

    estimate the area of the J in our 6X6 grid).

    2. In Fathom, create a collection with attributes x and y,

    using the formula random (5) for both attributes. 223. Create an attribute Shaded using the formula (x+y)<25.

    4. Generate 1000 cases and then create a scatter plot of y

    against x.

    5. Drag your attribute Shaded into the middle of the scatter

    plot. You should see the quarter-circle area in blue.

    6. Create a Summary Table showing the count() of the whole

    collection the count(shaded=true) and the ratio of

    count(shaded=true)/count(). Compare this ratio to the

    value for Pi/4.

Questions to ask yourself:

    ; Why does the x vs. y scatterplot fill a square of side-

    length 5?

    ; Why does the blue shaded area take the shape of a quarter

    circle?

    ; Why is the ratio in step 6 above close to the value of

    Pi/4?

    Investigating the area enclosed by a parabolic curve inscribed in a square.

    1. Create a new scatterplot of y against x.

    2. Add the following function to this scattterplot: y=(x-22.5)

    3. Create a new attribute (area_above) for the area above the

    parabola using your existing x and y attributes.

    4. Drag your area_above attribute into the middle of your new

    scatter plot. You should see the area above the parabola

    in blue.

    5. Create a summary table showing the count() of the whole

    collection, the count(area_above) and the ratio

    count(area_above)/count().

    6. Create a slider and label it a. Set a to 1.00 to start and 2use a to edit the function for the parabola: y=a(x-2.5).

    7. Double click on the slider and set its lower limit to 0.5

    and its upper limit to 1.5. Adjust the value of a using

    your slider until your function plot passes through the

    upper two corners of your square region -- points (0,5)

    and (5,5).

    8. Now edit your area_above formula to also use a so that the

    area is the area above your new function plot (see screen

    shot from Fathom below).

    What do you notice about the proportion of the area of the square that is above this new parabola?

For the really ambitious:

    Use your knowledge of integral calculus to compute the area bounded by the parabola and the top edge of the square. Does this calculation verify your experimental results?

    Can you generalize this result for a parabola inscribed in any square (i.e. passing through the two upper corners of the square with its vertex at the midpoint of the bottom of the square)?

    Can you generalize the result for any rectangle (i.e. passing through the two upper corners of the rectangle with its vertex at the midpoint of the bottom of the rectangle)?

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