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# Virtual Laboratory in Probability and Statistics

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TIMES 3 Technological Integrations in Mathematical Environments and Studies Jacksonville State University 2010 Virtual Laboratory in Probability and Statistics Google Go Virtual Lab UAH to the Applets Let s play Roulette 1 Find the mean for each bet 2 Find the standard deviation for each bet 3 What information is given by these statistics 4 If you had a limited amount of money but you wanted to play roulette for a long time what bet would accomplish your goal 5 Would your choice of bet make any difference on your net winnings in the long run 6 Choose any bet and run the simulation 500 times Based on Counting Permutation An ordered arrangement of all or part of a set of objects These can almost always be solved using the multiplication rule The number of arrangements of n things is n By Definition 0 1 n nPr the number of arrangements of n things taken r at a time is n r Examples Six photos can be arranged on a shelf ways The top two finishers in order in an 8 dog race can come in ways Exacta A Trifecta is the top 3 in order a Superfecta is the top 4 in order Count these Three different flavored scoops of ice cream can be arranged on a cone ways There are different possible 3 scoop cones using 31 different flavors if order matters Counting Combination An unordered selection of all or part of a set of objects There will always be fewer combinations than permutations given the same n and r nCr the number of ways to select r objects n n r r from n is given by Examples There are possible foursomes out of 7 golfers How many ways are there to fit 6 kids into a van if you don t care where they sit The top two finishers in an 8 dog race without regard to order can come in ways Quinella There are different possible 3 scoop cones using 31 different flavors is order doesn t matter Compare the years required to taste every 3 scoop cone using 31 flavors when order matters and when it doesn t Calculator Practice 7 14 Factorials are the fastest growing functions 7P0 7C0 7 7 C7 P7 P5 P6 12 5 C1 10 20 C8 The Binomial Distribution A binomial experiment has 1 A fixed number of trials n 2 Two outcomes success and failure 3 A constant probability of success p If there are only two outcomes and the probability of success is p then the probability of failure is 1 p 4 Independent trials The Binomial Random Variable X the number of successes out of n trials The value of X must be an integer between 0 and n inclusive Binomial Probability Given a binomial random variable x in a distribution with parameters n and p x P X n C x p 1 p n x Recall the example of tossing 3 coins and guessing the outcome Central Tendency and Variability The mean of the binomial distribution is given by np The standard deviation of the binomial distribution is given by n p 1 p At a carnival game you are offered the opportunity to bet one dollar on a number from one to six on a single roll of two dice If your number comes up on one die you win 2 If it comes up on both dice you win 5 Only if the number does not appear on either die do you If we think this one out as a binomial distribution there are n 2 trials in the die rolls Let s say a success is that we roll a one Then the probability of success is p 1 6 and the probability of failure is 5 6 The random variable x counts the number of successes x P x 0 2C0 1 6 0 5 6 2 694 1 2C1 1 6 1 5 6 1 278 2 2C2 1 6 2 5 6 0 028 We would come to the same probabilities by counting the 25 ways to lose and the 11 ways to win out of the 36 outcomes Now that we know the probabilities for rolling a one either once or twice or not at all we can define a random variable Y to see if the game is fair Y P Y 1 694 2 278 5 028 E x 0 SD 1 58 This game is fair Note that X is binomial and Y is not Why Chuck a Luck It costs 1 to play this game Pick a number from 1 to 6 The house rolls 3 fair dice If your number shows up once you win 1 If your number shows up twice you win 2 If your number shows up three times you win 3 So do you want to play Virtual Lab Pepys Problem In 1693 Samuel Pepys a member of the British Navy and later Parliament asked Isaac Newton whether it is more likely to get at least one ace in 6 rolls of a die or at least two aces in 12 rolls of a die This problems is known a Pepys problem naturally Pepys had fair dice in mind From the Virtual Lab home page go to the list of Applets Under the Dice Spinners heading select Dice Experiment With n 6 dice run the experiment 2000 times and find the binomial probability of rolling Z 1 ace on 6 dice Distribution gives the classical probability Data gives the empirical probability based on your simulation Remember to change the random variable from the default Y sum to Z number of aces P one ace on 6 dice Now with n 12 run the simulation 2000 times and find the binomial probability of rolling Z 2 aces on 12 dice Compare the results P two aces on 12 dice So what s the answer to Pepys question Calculate the following probabilities Use the formula or the applet Distribution values P 3 aces with 18 dice P 4 aces with 24 dice P 5 aces with 30 dice Make a conjecture that extends Pepys problem to more than 30 dice Questions Japanese Korean and Chinese dice represent the ace as a large red spot

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