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.
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Probability and StatisticsIntuitively AccessibleMathematically 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.
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.