Math215 Statistical Concepts Franklin University Coin Toss Project Part One: Subjective Probability (Your Guesses) Before tossing the coins, make the following predictions. Remember the rule of a probability distribution is that the sum of all possible mutually exclusive outcomes has to be one (1). Note that getting exactly 3 heads and getting exactly 5 heads are mutually exclusive outcomes. They cannot occur on the same toss but you are figuring probabilities for an infinite number of tosses. Also, think about how the probability of getting 3 heads relates to the probability of getting 7 tails. Think about the situation and then answer the questions below with your best guesses. Do not look up or compute these probabilities, but do think about the values. You will be graded on the reasonableness of your guesses. These are guesses, not computed values. Step 1 Answer the following questions. (You will need to include all of the answers to the questions below in your paper.) 1. What percent of the time do you expect to get 5 heads? ____________ 2. What percent of the time do you expect to get 3 heads? ____________ 3. ____________ What percent of the time do you expect to get 7 heads? 4. What percent of the time do you expect to get no heads? ____________ 5. What percent of the time do you expect to get all heads? ____________ 6. Use your guesses above to fill in the chart below for all of the outcomes. X = Heads Prob(X) 0 1 2 3 4 5 6 7 8 9 7. Adjust your guesses for 1-6 above until the two rules of probability are satisfied (Each value is between 0 and 1 and the sum of all of the probabilities is 1). 8. What is the expected value for heads after tossing 600 coins? 10 ____________ 9. How is the probability of getting three heads (# 2 above) related to the probability of getting three tails (# 3 above)? Explain your answer. _______________________________________________________________ 10. How is the probability of getting three heads related to the probability of getting seven tails? Explain your answer. ____________________________________________________________ _____________________________________________________________ 1190314 Math215 Statistical Concepts Franklin University Coin Toss Project Part Two: Empirical Probability and Setup the Excel Spreadsheet. In this part of this project, you will determine empirical probabilities and set up your Excel spreadsheet. Step 2 Toss the ten coins 60 times, each time recording the the number of heads for each throw in the \"Number of Heads\" column. You may use the application linked in the assignment for this. After 60 tosses you will be able to copy and paste your results into Excel. Step 3 Create an Excel spreadsheet with the following columns (See the Coin Toss Example.) You can fill some of these in now and the rest as you proceed through the project: A. Toss Number (number your tosses from 1 to 60 in this column.) B. Number of Heads (The results of step 2 above go in this column.) C. Cumulative Number of Heads (See step 4 below) D. Total Number of Coins Tossed so far (Number from 10 to 600 in increments of 10) E. Cumulative Percent Heads (See step 5 below) F. Leave this column blank G. Number of Successes (Number this column from 0 to 10) H. Subjective probability of Success (Put your guesses from the chart in Part One in this column. Remember these are probability numbers.) I. Actual Successes (These values are the counts of how many times you got 0, 1, 2, . . .10 heads). See step 6 below. J. Empirical Probability of Success (Each of the numbers in Column I divided by 60 tosses.) K. Computational probability of success (Instructions for completing this 2190314 Math215 Statistical Concepts column are in Part 3 of the Project.) Franklin University Step 4 Set up Excel to compute the cumulative number of heads (Column C). Use this column to have Excel compute the total number of heads that you have thrown so far. For example, if by the 15th throw you have gotten 60 heads total, then this column would read 60. If on the 16th throw you got 8 more heads the cumulative column would read 68 for the 16th throw. You can use cell referencing to do this.) Step 5 Set up Excel to compute the Cumulative Percent of Heads (Column E) by dividing each cell in the Cumulative Number of Heads column by the number in the Total Number of Coins Tossed cell. Format this column to show four decimal places. Step 6 Use Excel to count the number of tosses that resulted in zero heads, the number that resulted in one head, and the number that resulted in two heads, etc. (Column I). You may use the Excel \"=countif (data, X)\" function to do this. The \"data\" refers to the data in column B and \"X\" refers to the number of successes in column C. Step 7 Create a Graph Use Excel to create a line graph of the Cumulative Percent of Heads that uses the Toss Number as the X-axis. Format the Y-Axis to go from 0.400 to 0.600. You will copy and paste this graph into the body of your paper. 3190314 Math215 Statistical Concepts Franklin University Coin Toss Project Part Three: Classical Probability, Charts, and Analysis Paper In this part of the project, you will find the computational probabilities using Excel and create two more graphs and write up the formal paper. First note that tossing 10 coins (once) fits all of the requirements of a binomial distribution listed in your text: 1) There are 10 identical trials (each coin is one trial) so N = 10 2) Two outcomes are possible on each trial. Heads is a success. 3) The probability of success is the same on each coin so p = 0.5 4) Each coin is independent of the others. Step 8 (Column K) Use the binomial function of Excel to compute the computational probabilities of zero through 10 successes. (Question: Why is this zero through 10?). The Excel command is \"=binom.dist(X, N, p, FALSE). X is the values in column G, there were 10 coins, p is the probability of success on each toss, and FALSE means that this is not the Cumulative probability. Step 9 (Graph) Compare the results of your guesses from Part One to the computational probabilities. Create a double column graph of your subjective probabilities that you guessed in the table in Part One with the computational probability (Columns H and K). You may need to copy the values in these columns into a different location on your paper so they are side-by-side. Make sure the graph has labels and axes (the X-axis needs to go from zero to 10, not 1 to 11). 4190314 Math215 Statistical Concepts Franklin University Step 10 (Graph) Compare the results of empirical probability with the computational probability. Create a double column graph of the relative frequency probabilities versus the computational probabilities (Columns J and K). Make sure the graph has labels, axes (the X-axis needs to go from zero to 10, not 1 to 11). Step 11 Using Word, write a formal paper. This must be in the third person (no \"I,\" \"me,\" \"you\" in the paper). Watch grammar and spelling. Include the following: 1. Cover sheet 2. Introduction: a paragraph outlining the purpose of the experiment, the procedure and discussing the probabilities. (No more than a page). 3. A results section in which you a. Answer all of the questions in Part One (Subjective- Guesses) and include the guesses chart from Part One. b. Paste and discuss the three graphs (the line graph and two column graphs) with a thorough explanation of what each graph shows. You should cut and paste each graph separately into Word from your spreadsheet. 4. A conclusion where you summarize the results of the experiment, 5. Appendix A: A page telling what you have learned. (8 points) Tell your instructor what you learned while completing this project. You may use first person in this section. 6. Appendix B: Two charts from Excel. Columns A through E will constitute Chart 1 and columns G through K will form Chart 2. Do not copy all of the Excel spreadsheet, just the required columns. DO NOT SUBMIT your Excel spreadsheet to the Assignment drop box, only this Word document. 5190314