Assignment_Midterm_ApurvaMohapatra

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Apr 3, 2024

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Team Assignment - Probability and Type I/Type II Errors Submitted by-Apurva Mohapatra A. Theoretical vs. Empirical Probability – Coin Toss Experiment Doctor D tossed three coins - a quarter, a dime, and a nickel, at the same time, 25 times. Each coin, when tossed, landed as either heads or tails. 1. What was the sample space size for this experiment? The sample size for this experiment is 8, i.e., 2*2*2=8 The results of the 25 tosses are listed in the matrix below. 2. Record the probabilities for each combination in the matrix below. Show each as a fraction In column a. Record the theoretical probability of each outcome/combination In column b. Record the empirical probability of each combination. 3. Record the totals for the probabilities in the bottom row of the matrix. Possible Combination/ Result Quarter Dime Nickel a. Theoretical Probability of this Combination (show as a fraction) Actual Count for this Combination b. Empirical Probability of this Combination (show as a fraction) HHH 1/8 3 3/25 HHT 1/8 4 4/25 HTH 1/8 0 0 HTT 1/8 7 7/25 THH 1/8 1 1/25 THT 1/8 5 1/5 TTH 1/8 2 2/25 TTT 1/8 3 3/25 Total 1 25 1 3. Do the empirical probabilities (column a.) agree with the theoretical probabilities (column b.) for each combination? Explain why they are the same or different. No, Empirical probabilities and theoretical probabilities are different. Because theoretical probability is number of ways A can occur per sample size and Empirical probabilities is number of outcome upon number of observation. 1 of 4

B. Sample Space Size – All Possible Outcomes Assume the following rules are true for any telephone number in North America: a. A telephone number in North America consists of ten numbers o a three-digit area code, o followed by a three-digit exchange/prefix, o followed by a four-digit line/subscriber number (AAA-EEE-XXXX). b. The area code cannot start with a 0 or 1. c. The exchange cannot start with a 0 or 1. d. Other than the rules in b. and c. above, any digit 0-9 can be used in any of the ten places in a phone number. With the above rules in mind, 1. How many different seven-digit phone numbers (excluding the area code) can be formed? EEE-XXXX = 8*10*10*10*10*10*10= 8 million numbers. E= 8 because the exchange cannot start with a 0 or 1. 2. Can a city of 3 million people be served by a single area code? Explain Yes, 8 million is greater than 3 million. And for a area code there can be 8 million possible numbers can be formed. 3. How many area codes are possible? Area code is 3 digit and cannot start with a 0 or 1. 8*10*10=800 4. 4. How many 10-digit phone numbers are possible? AAA-EEE-XXXX 8*10*10*8*10*10*10*10*10= 6,400,000,000= 6.4 billion II. Not too long ago, a. An area code had to have a 0 or 1 in the middle digit. With this additional rule, 1. how many area codes were possible? AAA= 8*2*10= 160 area codes 2. How many 10-digit phone numbers were possible? AAA-EEE-XXXX = 8*2*10*8*10*10*10*10*10= 128,000,000 = 128 million phone numbers 2 of 4

C. At Least Once Suppose that 4% of the students at a particular college have the H1N1 virus. P(H1N1)= 4/100=0.04. 1. If a student gets together with 15 other college students over a period of time, what is the theoretical probability that at least one of those 15 students has the H1N1 virus? Show as a percentage with 3 decimal places. 1 - [P (not A in one trial)] n n = number of trails P(Not getting virus)= 1-P(not H1N1)^n = 1-(1-0.04)^15= 1- (0.96)^15 =1-0.5420=0.457914 = 45.791% 2.If a student gets together with 30 other students over a period of time, what is the theoretical probability that at least one of those 30 students has the H1N1 virus? Show as a percentage with 3 decimal places. P(Not getting H1N1)= 1-P(not H1N1)^n = 1-(1-0.04)^30 = 1- (0.96)^30 =1-0.293858 = 0.706142=70.614% D. Expected Value – Lottery You buy a lottery ticket. It costs $10. The table shows the possible outcomes. Result Amount Won Probability Grand Prize $250 0.025 Pretty Good Prize $75 0.100 Okay Prize $10 0.250 Sucker Prize $1 0.300 No Prize $0 0.325 1. Compute the expected value for this game. Don’t forget that a ticket costs $10. Expected value = -10*1 + 250*0.025 + 75*0.100 +10*0.250+1*0.300+0*0.325 =6.55 is probability of wining the lottery. You need to subtract $10 from the expected value because you’re spending $10. 2.Based on the expected value, should you play this game? Why or why not? Yes, because the expected value is greater than zero ,i.e., 6.55 3 of 4

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E. Type I-Type II Errors The numbers for a particular type of tumor are as follows:  5 in 100 tumors are Grade P  A blood test determining whether a tumor is Grade P is 80% accurate.  A positive blood test result indicates that the tumor is Grade P. A negative result indicates that the tumor is not Grade P.  The blood test is given to 12,000 people with tumors 1. Complete the matrix below…..it should be similar to the one in the Type I-Type II Error notes. a. Label the cells to indicate true positives, false positives, false negatives, true negatives, and all totals and percentages and Type I/Type II errors. b. Add the calculated values to each cell of the matrix. b. Calculate totals and include them in the matrix. Test Results Grade P Not Grade P Total Positive 480 2,280 0 (Type I Error) 2,760 Negative 120 (Type II Error) 9,120 9,240 Total 600 11,400 12,000 2280 people were tested positive for grade P but didn’t have tumor. 120 people tested for negative for grade P tumor. 2. What is the chance that a positive blood test really means that a patient’s tumor is Grade P? State this as a percentage with 3 decimal places Grade P(positive)/ Total (positive)= 480/2760 =0.1739130435 = 17.391% 3. Assume that blood test results were negative for a patient. What is the chance that the patient’s tumor is really Grade P? State this as a percentage with 3 decimal places Grade P(negative)/ Total (negative)=120/9240=0.01298701299=1.298% 4 of 4