This module allows students to apply concepts related to discrete distributions to a simple dice experiment. Students will compare empirical data and a theoretical distribution to determine if the game fits a discrete distribution. This experiment involves the concept of long-term probabilities.
Class Time:
Names:
Student learning outcomes:
The student will compare empirical data and a theoretical distribution to determine if a Tet gambling game fits a discrete distribution.
The student will demonstrate an understanding of long-term probabilities.
Supplies:
1 game “Lucky Dice” or 3 regular dice
For a detailed game description, refer
here . (The link goes to the beginning of Discrete Random Variables Homework. Please refer to Problem #14.)
Round relative frequencies and probabilities to four decimal places.
The procedure
The experiment procedure is to bet on one object. Then, roll 3 Lucky Dice and count the number of matches. The number of matches will decide your profit.
What is the theoretical probability of 1 die matching the object?
$\_\_\_\_\_\_\_\_\_$
Choose one object to place a bet on. Roll the 3 Lucky Dice. Count the number of matches.
Let
$X$ = number of matches. Theoretically,
$X~B(\_\_\_\_\_\_,\_\_\_\_\_\_)$
Let
$Y$ = profit per game.
Organize the data
In the chart below, fill in the
$Y$ value that corresponds to each
$X$ value. Next, record the number of matches picked for your class. Then, calculate the relative frequency.
Complete the table.
x
y
Frequency
Relative Frequency
0
1
2
3
Calculate the Following:
$\overline{x}=$
${s}_{x}=$
$\overline{y}=$
${s}_{y}=$
Explain what
$\overline{x}$ represents.
Explain what
$\overline{y}$ represents.
Based upon the experiment:
What was the average profit per game?
Did this represent an average win or loss per game?
How do you know? Answer in complete sentences.
Construct a histogram of the empirical data
Theoretical distribution
Build the theoretical PDF chart for
$X$ and
$Y$ based on the distribution from the section titled "The Procedure".
$x$
$y$
$P\left(x\right)=P\left(y\right)$
0
1
2
3
Calculate the following
${\mu}_{x}=$
${\sigma}_{x}=$
${\mu}_{y}=$
Explain what
${\mu}_{x}$ represents.
Explain what
${\mu}_{y}$ represents.
Based upon theory:
What was the expected profit per game?
Did the expected profit represent an average win or loss per game?
How do you know? Answer in complete sentences.
Construct a histogram of the theoretical distribution.
Use the data
Calculate the following (rounded to 4 decimal places):
$\text{RF}$ = relative frequency
Use the data from the section titled "Theoretical Distribution" here:
For questions 1. and 2., consider the graphs, the probabilities and relative frequencies, the means and the standard deviations.
Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences. (Note: these answers may vary and still be correct.)
Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions. (Note: these answers may vary and still be correct.)
Thinking about your answers to 1. and 2.,does it appear that the data fit the theoretical distribution? In 1 - 3 complete sentences, explain why or why not.
Suppose that the experiment had been repeated 500 times. Which table (from "Organize the Data" or "Theoretical Distribution") would you expect to change? Why? How might the table change?
Questions & Answers
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Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
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Damian
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Azam
Hello
Uday
I'm interested in Nanotube
Uday
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Prasenjit
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.