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What, A Green Jawbreaker??? Probability and Statistics Intuitively Accessible Mathematically Complex • • • • Directly influenced by our everyday lives Lotteries Prizes 500 year floods Research related to what to eat, environmental conditions • Polls and elections Probabilistic Thinking • Transition from ordinary language to more precise mathematical meanings. • Students need to reason probabilistically about data rather than simply to provide answers to questions. • Reasoning about the relationship between data and probability. Why Study Probability • Chance is all around us—we do not live in a totally deterministic world. • In your group talk about how probability is used in your world. Why Learn About Probability • • • • • • • • Insurance Disease/medical treatments Genetics Weather Athletics Gamble (responsibly) Doing or not doing homework Retirement/investments Uses of Probability • • • • • • • • • • Genetics Weather Medicine Insurance Disease and health care Gambling Investments and retirement Athletics Doing or not doing assignment Driving above the posted speed limit We Must Teach Probability • Do you purchase an item based on consumer reports or the experience of one friend? • We must make decisions in the face of uncertainty on a daily basis. • Life changing decision can have a foundation in probability and statistics. • Our goal must be to develop students who can make informed decisions. A Green Jawbreaker??? • In the sack on your table you have 10 jawbreakers. • Do Not Look or Open Sack Unless Directed • Your task, after collecting some data, will be to guess the number of each color in your sack. Define Probability • But before we begin in your group define probability. chance of success Pr obability total in sample space Let the Data Collection Begin • Have someone in your group reach in the sack without looking – Pull out a jawbreaker – Look at the color – Put the jawbreaker back in the sack – Record the observation on the flip chart. • Is your group ready to take an estimate (not guess) as to the contents? • Repeat the above procedure. Probability Concepts • Notation for probability of an event. – P(event) or P(A) = • P(event) = 0 • P(event) = 1 • Can the probability of an event ever be greater than 1? Why • 0 < P(event) < 1 More Data • Repeat the drawing procedure. • Repeat the drawing procedure. • Any surprises? Are you ready to estimate? Probability Thoughts • If you flip a coin 4 times and get 4 heads, what is the probability of getting heads on the 5th toss? • What if you flip a coin 15 times and get 15 heads? • If you flip a coin 100 times will you get 50 heads and 50 tails. • State the law of large numbers in your own words. More Probability • All events in an experiment must have an equal chance of occurring. Could we use coins in our sack? A clear sack? • The sum of the probabilities of all possible events will equal what? • P(NA) is the probability of A not occurring. This is called the complement of an event. • If the P(A) = ¼ what is the P(NA)? • Birthday problem example More Data • Repeat the drawing procedure. • Are you ready to guess now? • Repeat the drawing procedure one more time. • How many would you need to draw to have a good idea of what is in your sack? • To be 100% sure. • Nothing is certain in probability. Here We Go • If you correctly estimate the contents of your sack you will get a jawbreaker. • Full disclosure—I do not know how many are in each sack. Extension Activity • Fill three sacks with jaw breakers, 8 red and 12 blue, 12 red and 8 blue, 3 red and 17 blue. • Tell the students that you forgot which is which. • Take one of the sacks and draw one at a time and replace. Repeat several times and have them decide which mixture has been chosen. Finishing Up • Go to another sack and repeat the process. • Go to the last sack and repeat. • Allow them to change their guess and make sure that they justify their selections. Probability Questions • Which is most likely to happen – Get more than 7 heads out of 10 tosses – or more than 70 heads out of 100 tosses? • In a family of 4 children is it more likely to have BBBB or BBGG? Other Activities • • • • • Toss a tack, what could happen Toss a styrofoam cup. Spin a coin rather than toss This requires a large number of trials In Angel there is a file Green Jawbreaker which contain reference to this activity.