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Mathematics for Elementary Teachers 10ed [2013]2

CHA P TE R11PROBABILITYProbability in the Everyday WorldJules Frazier/Photodisc/Getty ImagesKenna Love/Photodisc/Getty ImagesOne very popular application of probability is the.famous “birthday problem.” Simply stated, in a group.of people, what is the probability of two people having.the same month and day of birth? Surprisingly, the probability of such matching birth dates is about 0.5 when.there are 23 people and almost 0.9 when there are 40.people. An interesting application of this problem is the.birthdays of the 43 American presidents through George.W. Bush: Presidents Polk and Harding were both born on.November 2.© Bettmann/CORBISLeemage/Universal Images Group/Getty ImagesPascal communicated with Fermat about the problem.and, remarkably, each solved the problem by different.means. Thus, in this correspondence, Pascal and Fermat.laid the foundations of probabilityNow, probability is recognized in many aspects of our.lives. For example, when you were conceived, you could.have had any of 8,388,608 different sets of characteristics based on 23 pairs of chromosomes. In school, if you.guess at random on a 10-item true/false test, there is only.about a 17% probability that you will answer 7 or more.questions correctly. In the manufacturing process, quality control is becoming the buzzword. Thus it is important to know the probability that certain parts will fail.when deciding to revamp a production process or offer.a warranty. In investments, advisers assign probabilitiesto future prices in an effort to decide among various.investment opportunities. Another important use of.probability is in actuarial science, which is used to determine insurance premiums. Probability also continues to.play a role in games of chance such as dice and cards.© Bettmann/CORBISIt is generally agreed that the science of probability began in the sixteenth century from the so-called.problem of the points. The problem is to determine the.division of the stakes of two equally skilled players when.a game of chance is interrupted before either player has.obtained the required number of points in order to winHowever, real progress on this subject began in 1654.when Chevalier de Mere, an experienced gambler whose.theoretical understanding of the problem did not match.his observations, approached the mathematician Blaise.Pascal (see the following illustration) for assistance.President James K. Polk.Born: November 2, 1795President Warren G. Harding.Born: November 2, 1865484c11.indd 4847/31/2013 9:04:10 AM.Problem-Solving.Strategies.1. Guess and Test.2. Draw a Picture.3. Use a Variable.4. Look for a Pattern.5. Make a List.6. Solve a Simpler.ProblemDo a Simulation.A simulation is a representation of an experiment using some appropriate objects.(slips of paper, dice, etc.) or perhaps a computer program. The purpose of a simulation is to run many replications of an experiment that may be difficult or impossible.to perform. As you will see, to solve the following Initial Problem, it is easier to simulate the problem than to perform the actual experiment many times by questioning.five strangers repeatedly.Initial Problem.At a party, a friend bets you that at least two people in a group of five strangers will.have the same astrological sign. Should you take the bet? Why or why not?7. Draw a Diagram.8. Use Direct.Reasoning.9. Use Indirect.Reasoning.10. Use Properties.of Numbers.11. Solve an.Equivalent.Problem.12. Work Backward.13. Use Cases.14. Solve an Equation.15. Look for a.Formula.16. Do a SimulationClues.The Do a Simulation strategy may be appropriate when.A problem involves a complicated probability experimentAn actual experiment is too difficult or impossible to performA problem has a repeatable process that can be done experimentallyFinding the actual answer requires techniques not previously developedA solution of this Initial Problem is on page 541.485c11.indd 4857/31/2013 9:04:10 AM.AUTHOR