Statistical Analysis – Binomial Investigations - Mathematics Assignment Help

Assignment Task:

Task:

A LEGO tower is to be constructed using only red and blue bricks. The bricks are each selected at random. In the source of bricks available, the probability of selecting a blue brick is 0.25. The tower can be built to any height. The first four rows of the tower could look like this:
Example 1 Example 2
R ? Row 1? R
B R ? Row 2 ? B R
R R B ? Row 3 ? B B R
B R B B  Row 4  R R B R
Part A: Predicting the Pattern of the LEGO Tower Marks
1. Investigate towers up to 10 rows high. Calculate the probability that in each row there are no blue bricks, exactly one blue brick, exactly two blue bricks or exactly three blue bricks.
Summarise your findings in a table similar to the model shown below.

Number of blue bricks 0 1 2 3
Row 1
Row 2
Row 3
…
Row 10

2. What pattern in the probabilities of exactly 0, 1, 2 or 3 blue bricks do you observe? 1
3. Consider building the tower to a height of rows. Write generalised rules for exactly 0, 1, 2 or 3 blue bricks in the 10th row. Generalise this to the th row.

4. (i) Create a spreadsheet to calculate the probability of various numbers of blue bricks in rows of different lengths up to Row 20. Label the columns with the number of blue bricks and the rows with the number of bricks in the row as in the table given above.

(ii) What effect does altering the probability of a cell being coloured blue have on the patterns or rules you have discovered? Investigate at least three different values for the probability that the brick is blue.

5. Research the term ‘Galton Board’ on the internet. For the LEGO towers created above, if a red brick represents a ball falling to the left, and blue brick represents a ball falling to the right, explain how the towers can be used to represent the probabilities of balls falling down a Galton Board. Explain your answer.

6. (i) Watch The Wall – She’s So incredibly Lucky!.
(ii) Consider the simplified situation in which at each level of the wall, the ball can only fall one cell to the left or the right on the next level down. Both of these options have equal probability.
There are 7 cells at the top of the wall and 15 at the base. The number of cells in each row increases from 7 to 15 and thereafter, the wall is rectangular. There are 20 rows in total.
In the last row of the wall the cells contain the value to be ‘won’. These are, in
order from left to right: $1, $50 000, $100, $100 000, $10, $200 000, $1,
$300 000, $1, $400 000, $10, $500 000, $100, $1 000 000 and $1.
Explore the probability that a person will ‘win’ $1 000 000 on the first ball they drop from a cell of their choice. Explain your strategy, providing all relevant reasoning and working. You may submit a spreadsheet as part of your answer.
 

Part B: Jury Selection Procedures
It is expected that the composition of any jury will reflect the demographics of the surrounding community. When it does not, then doubts about fairness of the jury selection process and legal challenges can arise. Although jurors are not selected solely by chance, comparing the actual jury to the composition that would occur if they were selected at random can tell lawyers whether there are grounds to investigate the fairness of a selection process.

In 1953 there was an historic case concerning jury selection. A jury in Fulton County, Georgia had convicted Avery, an African-American, of a serious felony. The facts of the jury selection were as follows:
 There were no African-Americans on the jury.
 There were 165 814 African-Americans in Fulton County, which had a population of 691 797.
 The list of 21 624 potential jurors had 1115 African-Americans.
 A jury pool of 60 people was selected from the list of 21 624 potential jurors, from which the final pool of 12 actual jurors was selected. The pool of 60 people had no African-Americans in it. Note: You may require an online high-precision calculator which displays more decimal places than a handheld calculator to compute some of these probabilities.
1. If 12 jurors were selected at random from the population of Fulton County, estimate the probability that there would be no African-Americans on the jury.

2. Can having only 1115 African-Americans on the list of 21 624 potential jurors be attributed to chance?

3. The jury pool of 60 people was selected from the list of 21 624 potential jurors. Can getting no African-Americans in the jury pool be attributed to chance?

4. The US Supreme Court overturned Avery’s conviction. Outline the statistical evidence that could have been used in the appeal case regarding jury selection.

Marking Guidelines
Part A (4 marks) Understanding, Fluency and Communication Mark
1. The probabilities are calculated correctly. 1
2. At least one pattern in the probabilities is identified. 1
3.  Correct rules for the 10th row are stated.
 The generalised rule for the th row is correct.

Part A (7 marks) Problem Solving, Reasoning and Justification
4. (i) A correct spreadsheet has been created.
(ii) There is evidence of exploration of the effect of three different values for the probability.

5. There is a clear explanation of how the LEGO towers and/or the associated spreadsheet could be used to model the Galton Board. 1
6.  A spreadsheet to model the Wall has been created. Note that this spreadsheet may contain some errors.
 The correct probabilities have been calculated for the first 14 rows at the top of the Wall.
 The correct probabilities have been calculated for the last rows at the top of the Wall.
 The probability of ‘winning’ $1 000 000 is supported with justification.

Part B (6 marks) Understanding, Fluency and Communication
1.  The correct binomial distribution has been identified and the parameters have been stated clearly.
 The probabilities are calculated correctly.

2.  The correct binomial distribution has been identified and the parameters have been stated clearly.
 The probabilities are calculated correctly using a high-precision calculator.

3.  The correct binomial distribution has been identified and the parameters have been stated clearly.
 The probabilities are calculated correctly.

Part B (4 marks) Problem Solving, Reasoning and Justification
2. Justification for a correct comment uses calculated probabilities. 1
3. Justification for a correct comment uses calculated probabilities. 1
4. The statistical evidence is summarised and there is justification for a proposed conclusion.

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