Probability

10 February 2014

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Probability

The study of probability dates back to the mid 17th century throughcorrespondence between two mathematicians, Pierre de Fermat andBlaise Pascal.

Here is a quote from Calculus, Volume II by Tom M. Apostol (2ndedition, John Wiley & Sons, 1969): “A gambler’s dispute in 1654 ledto the creation of a mathematical theory of probability by two famousFrench mathematicians, Blaise Pascal and Pierre de Fermat.

Antoine Gombaud, Chevalier de Mere, a French nobleman with aninterest in gaming and gambling questions, called Pascal’s attentionto an apparent contradiction concerning a popular dice game.Thegame consisted in throwing a pair of dice 24 times; the problem wasto decide whether or not to bet even money on the occurrence of atleast one double six during the 24 throws.

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A seemingly well-established gambling rule led de Mere to believe thatbetting on a double six in 24 throws would be profitable, but his owncalculations indicated just the opposite.

This problem and others posed by de Mere led to an exchange of lettersbetween Pascal and Fermat in which the fundamental principles ofprobability theory were formulated for the first time. Although a fewspecial problems on games of chance had been solved by some Italianmathematicians in the 15th and 16th centuries, no general theory wasdeveloped before this famous correspondence.”

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Simulation of de Mere’s Problem

Roll a pair of dice 24 times. If you roll double six at least once, then youwin. If you never get double six on any of the 24 rolls, you lose. Once youare done, submit

A Win

B Lose

The clicker software will show what percentage of the class wins.

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It turns out that the probability of losing is

(35/36)24

which is approximately .51, or about 51%.

Then the probability of winning is approximately .49, or slightly lessthan half. Thus, de Mere was right that it is a little less than evenmoney to bet on getting a double six in 24 rolls. Later on we will seehow to come up with this calculation.

Doing the computation above requires a scientific calculator. Thereare free websites that do the same thing. One is

http://web2.0calc.com/

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The reason for mentioning calculators now is that there will be times thatwe will need some sort of calculator. In particular, when we discussinterest rates near the end of the semester, we’ll need to do some fairlycomplex computations.

I don’t encourage you to buy a scientific calculator. Only get one if youthink you’ll have use of it outside this class. There are plenty of websitesthat do calculations, including financial calculations. The calculatorprogram on a computer will do the same thing, so you’ll be able to do anycalculation we’ll need.

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If you have a smart phone you may already have a scientific calculator. Forexample, the iPhone comes with a calculator app. Rotating the phonetoggles between a regular and scientific calculator.

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Simulating de Mere’s Problem with Excel

We will use the Microsoft Excel spreadsheet deMereProblem.xlsx tosimulate our experiment. The advantage is that we can conduct manytrials in a short amount of time; many more than we could do by actuallyrolling dice.

This spreadsheet, and all others we will use, will be on the course website.

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Theoretical and Experimental Probability

This week we will explore some of the ideas that are used in thiscalculation. But first, we will focus on getting a better understandingof what is the meaning of probability.

If you conduct an experiment, such as the dice rolling we did earlier,you can compute an experimental probability, as we did. Ourpercentage of wins for the entire class is an experimental probabilityfor winning that game.

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If we say the probability of getting heads when you flip a coin is 50%,this is a theoretical probability. It is telling us what we expect to getif we flip a bunch of coins.

The reality is a little more complicated, as we’ll explore. Roughly, tosay that the probability (or chance) of flipping a coin and gettingheads is 50%, or .5, then on average, if you flip a coin many times,you will expect 50% of the flips being heads. However, what happensin a given set of flips can be nearly anything.

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The theoretical probability allows us to estimate what will happenwhen we conduct an experiment. However, we can use anexperimental probability to estimate theoretical probability. Therefore,each can be used to estimate the other.

For example, casinos can estimate how much money they will makefrom a game by knowing the theoretical probability of winning thatgame. What actually happens on a given day is an experimentalprobability. Because they have so many people playing, theirestimates are usually pretty good.

There is always a chance that somebody wins big on a given day.But, if they estimate income for longer periods of time, they’ll beeven more accurate.

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A Coin Flip Experiment

Flip a coin once and record the number of heads (either 0 or 1) with yourclicker.

We will show the class data.

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Another Coin Flip Experiment

Flip a coin 20 times and enter the number of heads you got.

We will tabulate the data for the entire class. What does the data indicateto you about the probability of getting a heads on a flip of a coin?

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Simulating the Coin Flip Experiment with Excel

We will use the spreadsheet Coin Flip.xlsx to simulate flipping a coin.

We will simulate flipping different numbers of coins with thespreadsheet. One thing to think about is how do you think thenumber of coins we flip will affect the results. How did flipping onecoin versus 20 coins affect our results in the previous experiment?

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Observations from the Simulation

How did the variability of the percentage of heads change as weincreased the number of flips?

The percentage of heads didn’t change as much from trial to trialwhen we flipped more coins. We are therefore getting a betterestimate of the theoretical probability the more coins we flip.

Do you think the spreadsheet confirmed that the probability is 50% toget a heads (assuming the spreadsheet does correctly simulate flippinga coin)?

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