ࡱ>  "bjbjcTcT 46>> +$===QQQ8lQ u`     !!!tttttttw"zt=!!"!!!t  st%%%! = t%!t%%~o8Tt LQ.%rXtt0 u;sTz%ztz=t !!%!!!!!tt%!!! u!!!!z!!!!!!!!! : Quiz or no quizIntroduction: This activity involves students exploring the outcomes of tossing a coin repeatedly to decide whether there is a quiz or not. Problem The teacher will make a deal with the students: Start with a counter at the top. Toss a coin four times. Heads means move the counter down and left. Tails means move the counter down and right. Cups 1, 2, or 5 = No Quiz Cups 3 or 4 = Quiz Is this a fair deal? Students predict how many times the counter will end up in each cup if the experiment is done 100 times.Plan  Toss the coin 4 times and follow the path to see which cup the counter ends up in. 91 Do the experiment 16 times and record the number of times the counter ends up in cup1, cup2, cup3, cup4 and cup5, therefore find the experimentally estimated probabilities that it ends up in cup1, cup2, cup3, cup4, and cup5. Combine the results with other students and find the experimentally estimated probabilities. Check: Is this a fair deal? How close was their prediction?  Analysis  Lead students to calculate the theoretical probabilities to verify their experimentally estimated results Calculate the theoretical probability that it ends up in cups 1, 2, or 5 (OR/PLUS concept or by counting all the outcomes ) Calculate the theoretical probability that it ends up in cups 3 or 4 (OR/PLUS concept) Sytematically list all the possible outcomes (the sample space) for landing in each cup. List the outcomes of the four tosses that end up in cup1: HHHH ----4 heads Find the number of ways to get to this cup1 (4 heads)---- 1 way List the outcomes of the four tosses that end up in cup2: HHHT, HHTH, HTHH etc ----3 heads 1 tail Find the number of ways to get to this cup2 (3 heads and 1 tail)-----4 ways List the outcomes of the four tosses that end up in cup3----2 heads 2 tails Find the number of ways to get to this cup3 (2 heads and 2 tails)-----6 ways List the outcomes of the four tosses that end up in cup4----1 head 3 tails Find the number of ways to get to this cup4 (1 head and 3 tails)------4 ways List the outcomes of the four tosses that end up in cup5----4 tails Find the number of ways to get to this cup5 (4 tails)-----1 way Introduce Pascal Triangle to find the number of ways to get to a particular combination Use the concept of AND/Independence/Multiplication to calculate (a) the theoretical probability of getting 4 heads---1/16 (b) the theoretical probability of getting 3 heads and 1 tail -----4/16 (c) the theoretical probability of getting 2 heads and 2 tails----6/16 (d) the theoretical probability of getting 1 head and 3 tails -----4/16 (e) the theoretical probability of getting 4 tails ------1/16 x 1 2 3 4 5 Total P(x) 1/16 4/16 6/16 4/16 1/16 1 Conclusion This is also an informal introduction to binomial distribution Calculate the theoretical probabilities of getting into cups 1, 2, or 5 Calculate the theoretical probabilities of getting into cups 3 or 4 Compare the experimentally estimated probabilities with the theoretical probabilities, the experimentally estimated probabilities should approach the theoretical probabilities as the sample size increases. ReflectionExtension activityStudetns construct a version of the game they think is fair.     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