# discrete and continuous probability paper

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**Option #1:** **Batting**

The batting average of a baseball player is the number of “hits” divided by the number of “at-bats.” Recently, a certain major league player’s at-bats and corresponding hits were recorded for 200 consecutive games. The consecutive games span more than one season. Since each game is different, the number of at-bats and hits both vary. For this particular player, there were from zero to five at-bats. Thus, one can sort the 200 games into six categories:

0 at-bats

1 at-bat

2 at-bats

3 at-bats

4 at-bats

5 at-bats

Consider the games where the player had exactly four at-bats. A similar analysis can be done for each of the other at-bats category. Download the file titled Bats. It contains a scatter plot of the four at-bats number of hits versus frequency. To compare the results to the Binomial Distribution, complete the following:

- Explain why the four at-bats is a binomial experiment.
- Using the Bats scatter plot, construct a frequency distribution for the number of hits.
- Compute the mean number of hits. The formula for the mean is .

Here, xi represent no. of hits (0, 1, 2, 3, 4) and*f*i is the corresponding frequency. Explain what the numerical result means. - From the frequency distribution, construct the corresponding probability distribution. Explain why it is a probability distribution. Then, use Excel to make a scatter plot of the probability distribution:

Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting). - Using the frequency distribution, what is the player’s batting average for four at-bats? In part 3, note that the numerator in the formula for the mean is the total number of hits. The total number of at-bats is the denominator of the formula for the mean multiplied by 4.
- The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of “success” on each trial. Using Excel, construct the Binomial Probability Distribution for four trials, n, and probability of success, p, as the batting average in part 5. Here is an explanation of the BINOM.DIST function in Excel.

For example, In Excel

=BINOM.DIST(7,15,0.7, FALSE)

represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p. - Using the formula for the mean of the binomial distribution, what is the mean number of successes in part 6 up above?
- In Excel, make a scatter plot for the binomial distribution. The instructions for making one are in part 4 up above.
- Use the results up above to compare the probability distribution of four at bats and the Binomial Distribution. Compare the means in parts 4 and 6, too. If the probability distribution of 4 at bats and the Binomial Distribution differ, explain why that is so.

Write a report that adheres to the Written Assignment Requirements under the heading “Expectations for CSU-Global Written Assignments” found in the CSU-Global Guide to Writing and APA Requirements*.* As with all written assignments at CSU-Global, you should have in-text citations and a reference page. An example paper is provided in the *MTH410 Guide to Writing with Statistics.*

Submit your Excel file in addition to your report.

Requirements:

- Paper must be written in third person.
- Your paper should be four to five pages in length (counting the title page and references page) and cite and integrate at least one credible outside source. The CSU-Global Library is a great place to find resources.
- Include a title page, introduction, body, conclusion, and a reference page.
- The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society as a whole, or it might discuss or describe the unique terminology associated with the topic.
- The body of your paper should answer the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
- The conclusion should summarize your thoughts about what you have determined from the data and your analysis, often with a broader personal or societal perspective in mind. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
- Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment.
- Document formatting, citations, and style should conform to the CSU-Global Virtual Library CSU-Global Guide to Writing and APA: Introduction. A short summary containing much that you need to know about paper formatting, citations, and references is contained in the New Sample APA Paper. In addition, information in the CSU-Global Virtual Library under the Writing Center/APA Resources tab has many helpful areas (Writing Center, Writing Tips, Template & Examples/Papers & Essays, and others).

**Option #2:** **Hockey**

The success average of a hockey player is the number of “points scored” divided by the number of “shots on goal.” Recently, a certain professional league player’s shots on goal and corresponding points scored were recorded for 400 consecutive games. The consecutive games span more than one season. Since each game is different, the number of shots and points scored both vary. For this particular player, there were from 0 to 15 shots. Thus, one can sort the more than 400 games into 16 categories:

0 shots

1 shot

2 shots

.

.

.

14 shots

15 shots

Consider the games where the player had exactly five shots on goal. A similar analysis can be done for each of the other shots category. Download the file titled Hockey. It contains a scatter plot of the Five Shots number of successes versus frequency. To compare the results to the Binomial Distribution, complete the following:

- Explain why the five shots is a binomial experiment.
- Using the Hockey scatter plot, construct a frequency distribution for the number of successes.
- Compute the mean number of successes. The formula for the mean is .

Here, xi represent no. of successes (0, 1, 2, 3, 4, 5) and*f*i is the corresponding frequency. Explain what the numerical result means. - From the frequency distribution, construct the corresponding probability distribution. Explain why it is a probability distribution. Then, use Excel to make a scatter plot of the probability distribution:

Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting). - Using the frequency distribution, what is the player’s success average for five shots? In part 3, note that the numerator in the formula for the mean is the total number of successes. The total number of shots is the denominator of the formula for the mean multiplied by 5.
- The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of “success” on each trial. Using Excel, construct the Binomial Probability Distribution for five trials, n, and probability of success, p, as the success average in part 5. Here is an explanation of the BINOM.DIST function in Excel

For example, In Excel

=BINOM.DIST(7,15,0.7,FALSE)

represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p. - Using the formula for the mean of the binomial distribution, what is the mean number of successes in part 6 up above?
- In Excel, make a scatter plot for the binomial distribution. The instructions for making one are in part 4 up above.
- Use the results up above to compare the probability distribution of five shots and the Binomial Distribution. Compare the means in parts 4 and 6, too. If the probability distribution of five shots and the Binomial Distribution differ, explain why that is so.

Write a report that adheres to the Written Assignment Requirements under the heading “Expectations for CSU-Global Written Assignments” found in the CSU-Global Guide to Writing and APA Requirements*.* As with all written assignments at CSU-Global, you should have in-text citations and a reference page. An example paper is provided in the *MTH410 Guide to Writing with Statistics.*

Submit your Excel file in addition to your report.

Requirements:

- Paper must be written in third person.
- Your paper should be four to five pages in length (counting the title page and references page) and cite and integrate at least one credible outside source. The CSU-Global Library is a great place to find resources.
- Include a title page, introduction, body, conclusion, and a reference page.
- The introduction should describe or summarize the topic or problem. It might discuss the importance of the topic or how it affects you or society as a whole, or it might discuss or describe the unique terminology associated with the topic.
- The body of your paper should answer the questions posed in the problem. Explain how you approached and answered the question or solved the problem, and, for each question, show all steps involved. Be sure this is in paragraph format, not numbered answers like a homework assignment.
- The conclusion should summarize your thoughts about what you have determined from the data and your analysis, often with a broader personal or societal perspective in mind. Nothing new should be introduced in the conclusion that was not previously discussed in the body paragraphs.
- Include any tables of data or calculations, calculated values, and/or graphs associated with this problem in the body of your assignment.
- Document formatting, citations, and style should conform to the CSU-Global Virtual Library CSU-Global Guide to Writing and APA: Introduction. A short summary containing much that you need to know about paper formatting, citations, and references is contained in the New Sample APA Paper. In addition, information in the CSU-Global Virtual Library under the Writing Center/APA Resources tab has many helpful areas (Writing Center, Writing Tips, Template & Examples/Papers & Essays, and others).