KH -1308 SIP Activity 2-Analysis of Coin Toss-AOP Disc

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1308

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Feb 20, 2024

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SIP Activity 2: Coin Toss Analysis Coin Tosses (Part 1) In this activity each student (groupmate) will first toss a coin 10 times (or access the StatCrunch applet* to simulate 10 tosses) and repeat the process to produce 4 sets of 10 tosses each . *StatCrunch applet is found via QUICK LINKS > MyLab StatCrunch > StatCrunch website > Interactive applets > Simulations > “Probability of a head with a fair coin.” You may select 5 flips at a time in the applet . Record your results in the table below. The exact sequence of heads and tails should be written in the Outcomes columns. Set # Outcomes (Head or Tail) No. of Heads 1 H T H T H T T T T T 3 2 H H H H T T H H T T 6 3 H T T T H T H H H H 6 4 H H T H H T T T T T 4 FIRST POST : Each groupmate must post the results of their four sets in the Discussion board. After all groupmates have posted their results, compile the data for “No. of Heads” of the group in the table below. For example , if there are 5 groupmates, there should be 20 sets; add rows as needed if you have more. Make a histogram of the ‘Frequency of sets resulting in ___ number of heads ,’ i.e., the frequency with which a certain number of heads occurred: 0-time, 1 time, 2 times, 3 times, … etc., among all the group’s sets. Your histogram can be created with StatCrunch or Excel and inserted into this document. Or use the blank graph provided on the next page to draw your histogram. SECOND POST (and subsequent posts) : Share your histogram of the group’s results, and your answers to all remaining questions and tables of SIP Activity 2, in the Discussion board. Page 1 of 4 Frequency of sets resulting in a certain no. of heads Set # No. of Heads 1 3 2 6 3 6 4 4 5 4 6 4 7 2 8 7 9 4 10 5 11 3 12 4

Empirical Probabilities of Coin Tosses Consider the group’s relative frequencies of obtaining a certain number of heads as empirical probabilities. So, based on all the group’s sets , calculate the following empirical probabilities. 1. What is the empirical probability of obtaining 0 heads in 10 tosses? P (obtaining 0 heads) is 0 2. What is the empirical probability of obtaining 1 head in 10 tosses? P (obtaining 1 heads) is 0 3. What is the empirical probability of obtaining 9 heads in 10 tosses? P (obtaining 9 heads) is 0 4. What is the empirical probability of obtaining 10 heads in 10 tosses? P (obtaining 10 heads) is 0 5. Based on your group’s empirical probabilities, were any of the events, 0, 1, 9, or 10 heads, considered unusual events? Why or why not? Yes, they were all less than 0.5%. Theoretical Probabilities of Coin Tosses (Part 2) Think of tossing a coin 10 times as a binomial probability experiment, where the probability of heads is p = 0.5 . Compute the remaining theoretical probabilities of obtaining a certain number of heads in 10 tosses in the table below.* Enter your earlier empirical results in the last two columns. ( Ex : the empirical probability of obtaining exactly 3 heads is the proportion of all your group’s sets that resulted in obtaining exactly 3 heads.) No. of heads No. of combinations Probability of an individual outcome Theoretical Probability of Event, “No. of heads.” Frequency of sets with this No. of heads Empirical Probability of this No. of heads 0 10 ! 0 ! ( 10 − 0 ) ! = 1 1 2 1 × ( 1 2 ) 0 × ( 1 − 1 2 ) 10 = 0.001 0 0/10=0 1 10 ! 1 ! ( 10 − 1 ) ! = 10 1 2 10 × ( 1 2 ) 1 × ( 1 − 1 2 ) 9 = 0.01 0 0/10=0 2 10 ! 2 ! ( 10 − 2 ) ! = 45 1 2 45 ( 1 2 ) 2 ( 1 − 1 2 ) 8 = 0.044 1 1/10=0.1 3 10 ! 3 ! ( 10 − 3 ) ! = 120 1 2 120 ( 1 2 ) 3 ( 1 − 1 2 ) 7 = 0.117 2 2/10=0.2 Page 2 of 4

4 10 ! 4 ! ( 10 − 4 ) ! = ¿ 1 2 0.205 5 5/10=0.5 5 10 ! 5 ! ( 10 − 5 ) ! = ¿ 1 2 0.246 0 0/10=0 6 10 ! 6 ! ( 10 − 6 ) ! = ¿ 1 2 0.205 2 2/10=0.2 7 10 ! 7 ! ( 10 − 7 ) ! = ¿ 1 2 0.117 1 1/10=0.1 8 10 ! 8 ! ( 10 − 8 ) ! = ¿ 1 2 0.044 0 0/10=0 9 10 ! 9 ! ( 10 − 9 ) ! = ¿ 1 2 0.01 0 0/10=0 10 10 ! 10 ! ( 10 − 10 ) ! = ¿ 1 2 0.001 0 0/10=0 *Hint: You may use StatCrunch > Stat > Calculators > Binomial to quickly calculate the Theoretical Probabilities. Theoretical vs. Empirical The theoretical probabilities of obtaining a certain number of heads in 10 tosses of a coin were calculated using the binomial probability function. 1. Would it be unusual, theoretically, to observe 0, 1, 9, or 10 heads in 10 tosses of a coin? Why or why not? (Hint: What are the probabilities of these events?) Yes as the probability of these events is less than 0.5 2. Discuss the differences, if any, between the theoretical probabilities in part 2 and your empirical probabilities in part 1. Do differences indicate that the coins were not fair coins? What may be done to narrow the gap between theoretical and empirical probabilities? 1. Theoretical Probability (Column 4): These are the probabilities calculated based on the theoretical probability formula for a binomial distribution with a fair coin (in this case, 1/2 for each head or tail). 2. Empirical Probability (Column 6): These are the probabilities calculated based on the actual outcomes observed in our experiment (the number of sets with a specific number of heads divided by the total number of sets).  Differences: The differences between theoretical and empirical probabilities may arise due to randomness or experimental error. In our case, there are some differences, but they are relatively small. Page 3 of 4

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 Fairness of Coins: Small differences alone may not necessarily indicate that the coins are unfair. Randomness and variation are inherent in probability experiments. However, if there were consistently significant differences, it might raise questions about the fairness of the coins or the experimental setup.  Closing the Gap: To narrow the gap between theoretical and empirical probabilities, you can increase the number of trials (coin tosses). As the number of trials increases, the empirical probabilities tend to converge toward the theoretical probabilities. This is known as the law of large numbers. In summary, while small differences are expected due to randomness, if you have concerns about the fairness of the coins, you may want to conduct more trials to see if the empirical probabilities converge towards the theoretical values. If significant disparities persist, further investigation into the coin or experimental setup may be warranted. Page 4 of 4