1 Random Experiments from Random Experiments

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1 Random Exeriments from Random Exeriments. Bernoulli Trials The simlest tye of random exeriment is called a Bernoulli trial. A Bernoulli trial is a random exeriment that has only two ossible outcomes: success or failure. If you i a coin and want it to come u heads, this is a Bernoulli trial where heads is success, and tails is failure. If you roll two dice on a monooly board, and you want it to come u doubles to get you out of jail, this is a Bernoulli trial Success is rolling doubles; everything else is a failure. If one of the dice gets hung u on the Chance Deck, that s a "do-over." The only outcomes that count are success or failure: doubles or not. If you are laying Rock-Paer-Scissors, and want to win, then your rock will break your oonent s scissors; your scissors will cut your oonent s aer, and your aer will cover your oonent s rock. All these are successes. But your oonent s rock will break your scissors; their scissors will cut your aer, and their aer will cover your rock. All these are failures. The other ossibilities are both rocks, both scissors or both aer; none of these count, they all lead to "do-overs." So under the familiar rules, Rock-Paer-Scissors is a Bernoulli trial. There is no way to redict the result of an individual instance of a Bernoulli trial before it occurs. (Unless of course, the Bernoulli trial is such that one outcome is actually not ossible.) Assuming that both success and failure are ossible, to be a random trial, the result of an individual try must be unredictable. Here are some examles of Bernoulli trails, and the robability models that seems to re ect them best:.. Fair Coin Toss: A fair coin is tossed, and one side is chosen as a success. The coin is tossed once: Success Failure.. A Fair Die Toss A fair six sided die is tossed. Each face of the die has a number from to. One of those numbers is declared a success. The die is rolled once: Success..3 A Ball is Drawn From an Urn Failure An urn is lled with n balls, and exactly m of those balls are of a seci c color. (For some reason it s always an urn in advanced math courses. Otherwise it

2 might just be a box, or a deck of cards, or any other drawing-from-a-grou-andlooking-for-one-tye kind of exeriment.) One ball is drawn, without looking, from the urn: Success Failure m n..4 Two fair dice are thrown Two fair six sided dice are thrown. Each face of the die has a number from to. One can call an odd sum a success, but then the acceted robability model is Success Failure Unless you are unsure of the fairness of the dice or the toss, it would do just as well to i a coin. One could call doubles a success, but then the acceted robability model is m n Success Failure Again unless you are unsure of the fairness of the dice or the toss, it would do just as well to roll one die. One could call doubles a failure and not doubles a success and obtain Success Failure But rolling one die would do just the same. One could declare one of the ossible totals that come u a success. Here the robability model will deend on the number, but the various ossibilities are k Success Failure 3 or or or 8 or or Now 7 is a oular sum to make a success, but mathematically one might just as well just roll one fair die. One could call a combination of totals a success. With the information above one can comute the correct robability of success. One oular air to declare a success is f7; g: Then we have 9 Success Failure 7 9

3 .. The Generic Bernoulli Trial In this exeriment we assume we have a Bernoulli trial with a known robability of success. The model is simly Success Failure :. Using Bernoulli Trials to Create Larger Exeriments In almost every situation where we are looking at a random event as a Bernoulli trial, we are thinking of reeating the trial a large number of times. We want to create a larger exeriment by reeating the trial a large number of times, and counting of the number of successes. We will start small and work out way u. Suose we toss a fair die and consider a to be a success. First we create an exeriment where we toss a fair die twice, and set a samle sace that kees track of the outcome of every toss: fss; SF; F S; F F g: Of course SS means two successes, SF means a success followed by a failure, F S is a failure and a success, and F F is two failures. Now we clearly want to assume that the tosses are indeendent. Thus the outcome of the rst toss and the second toss are mathematically indeendent. We can comute the robabilities of these events by thinking of them as two searate indeendent events. We will denote the robability of the larger exeriment by. Thus (SS) = (S)(S) = = 3 (SF ) = (S)(F ) = = 3 (F S) = (F )(S) = = 3 (F F ) = (F )(F ) = = 3 Of course, we want a robability model for the number of successes. events we are interested in are Success = ff F g Success = fsf; F Sg Success = fssg Now the We comute these, but include a row that shows where the robabilities came from # Success (#) (#)

4 Next we try an exeriment based on three die tosses. The samle sace that kees track of the outcome of every toss is fsss; SSF; SF S; SF F; F SS; F SF; F F S; F F F g: The outcomes of the rst toss, the second toss, and the third toss are assumed to be mathematically indeendent. We can comute the robabilities of these events by thinking of them as three searate indeendent events. We will denote the robability of the larger exeriment by 3. Thus 3 (SSS) = (S)(S)(S) = = 3 (SSF ) = (S)(S)(F ) = = 3 (SF S) = (S)(F )(S) = = 3 (SF F ) = (S)(F )(F ) = = 3 (F SS) = (F )(S)(S) = = 3 (F SF ) = (F )(S)(F ) = = 3 (F F S) = (F )(F )(S) = = 3 (F F F ) = (F )(F )(F ) = = The events we are interested in are Success = ff F F g Success = fsf F; F SF; F F Sg Success = fssf; SF S; F SSg 3 Success = fsssg We notice that the robabilities of the samle oints that make u each one of these events are all the same: 3 3 (F F F ) = 3 (SF F ) = (F SF ) = (F F S) = 3 (SSF ) = (SF S) = (F SS) = 3 (SSS) = 3 4

5 We have written these robabilities this odd way to illustrate the strong attern they exhibit. The robability of any one of these samle oints is just to a ower given by the number of S s times to a ower given by the number of F s. Thus the robability model we are searching for is # Success 3 (#) ( ) ( )3 3 ( ) ( )3 3 ( ) ( )3 ( ) ( )3 (#) 7 Of course, the two threes in this come from the fact that both fsf F; F SF; F F Sg and fssf; SF S; F SSg have three entries. Next we try moving to an exeriment based on four coin tosses. A few things come quickly, the samle sace will be twice as big 7 fssss; SSSF; SSF S; SSF F; SF SS; SF SF; SF F S; SF F F F SSS; F SSF; F SF S; F SF F; F F SS; F F SF; F F F S; F F F F g: The outcomes of the four tosses will be assumed to be mathematically indeendent. We can comute the robabilities of these events by counting the number of successes and failures. We will denote the robability of the larger exeriment by 4. By examle 4 4 (SSSS) = 3 4 (SSF S) = 4 (SF F S) = 3 4 (SF F F ) = 4 4 (F F F F ) = The events we are interested in are Success = ff F F F g Success = fsf F F; F SF F; F F SF; F F F Sg Success = fssf F; SF SF; SF F S; F F SS; F SF S; F SSF g 3 Success = fsssf; SSF S; SF SS; F SSSg 4 Success = fssssg Again the robabilities of the samle oints that make u each one of these events are all the same. Thus the robability model we are searching for is # Success (#) ( ) ( )4 4 ( ) ( )3 ( ) ( ) 4 ( )3 ( ) ( )4 ( )

6 If we try to move on to ; or even n reeated tosses, we can redict the form of the result we will get # Success k = ; ; ; 3 : : : n n (#) np k ( )k ( )n k where n P k is a natural number that counts the number of di erent ways k S s and (n k) F s can be lined u. Here are some charts that illustrate the robability models for various n..8 n= =/ n=3 =/ n= =/

7 .3 n= =/ n= =/ Notice that as the number of trials gets larger, the robabilities of the larger number of success dros o to almost nothing. If we take even more trials and draw those charts, we might as well leave those very unlikely cases o the chart to make it result clearer. The n = might be better resented as. n= =/ For n = 3 7

8 n=3 =/ After a while, the robabilities that there are too few success becomes very small,too small to kee on the chart. We simly leave them o.. n=3 =/ Now all the scales in these charts are changing, but those changes do illustrate how regular these robability models become. It takes more than a coule of trials, but by the time we get to trials, the robability is beginning to take on a fairly regular shae. The most remarkable thing about this emerging shae is that it aears in every other exeriment based on the reetition of a Bernoulli trial. Suose we i a fair coin instead of tossing a die. We will consider a heads to be a success. First we create an exeriment where we i the coin twice, and set a samle sace that kees track of the outcome of every i: fss; SF; F S; F F g: We assume that the outcome of the rst toss and the second toss are mathematically indeendent. We can comute the robabilities of the larger exeriment 8

9 and denote the result by. Thus (SS) = (S)(S) = = 4 (SF ) = (S)(F ) = = 4 (F S) = (F )(S) = = 4 (F F ) = (F )(F ) = = 4 Of course, we want a robability model for the number of successes. events we are interested in are Now the Success = ff F g Success = fsf; F Sg Success = fssg We comute these, but include a row that shows where the robabilities came from # Success (#) (#) 4 Next we try an exeriment based on three coin is. The samle sace that kees track of the outcome of every is is fsss; SSF; SF S; SF F; F SS; F SF; F F S; F F F g: The outcomes of the rst toss, the second toss, and the third toss are assumed to be mathematically indeendent. We will denote the robability of the larger exeriment by 3. Thus 4 3 (SSS) = (S)(S)(S) = = 8 3 (SSF ) = (S)(S)(F ) = = 8 3 (SF S) = (S)(F )(S) = = 8 3 (SF F ) = (S)(F )(F ) = = 8 3 (F SS) = (F )(S)(S) = = 8 3 (F SF ) = (F )(S)(F ) = = 8 3 (F F S) = (F )(F )(S) = = 8 3 (F F F ) = (F )(F )(F ) = = 8 9

10 The events we are interested in are Success = ff F F g Success = fsf F; F SF; F F Sg Success = fssf; SF S; F SSg 3 Success = fsssg We notice that the robabilities of the samle oints that make u each one of these events are all the same: 3 3 (F F F ) = 3 (SF F ) = (F SF ) = (F F S) = 3 (SSF ) = (SF S) = (F SS) = 3 (SSS) = 3 We have written these robabilities this odd way to illustrate the resemblance to the dies toss examle. The robability of any one of these samle oints is just to a ower given by the number of S s times to a ower given by the number of F s. Because the robability of success is the same as the robability of failure, the resulting roduct is always 3 (F F F ) = 3 (SF F ) = (F SF ) = (F F S) 3 (SSF ) = (SF S) = (F SS) = 3 (SSS) = Thus the robability model we are searching for is 3 = 8 # Success 3 (#) ( ) ( )3 3 ( ) ( )3 3 ( ) ( )3 ( ) ( )3 3 3 (#) Of course, the two threes in this come from the fact that both fsf F; F SF; F F Sg and fssf; SF S; F SSg have three entries. By now some things are quite obvious. Whether we toss a die, i a coin, or do any Bernoulli trial, the samle sace for n reeats will be the same. For n = 4; it will be fssss; SSSF; SSF S; SSF F; SF SS; SF SF; SF F S; SF F F F SSS; F SSF; F SF S; F SF F; F F SS; F F SF; F F F S; F F F F g: The outcomes of the four reeats of the trial will be assumed to be mathematically indeendent. We can comute the robabilities of these events by

11 counting the number of successes and failures. If the robability of success is in one trial, then the robability of failure is ( ). For a fair coin i, the are both : In the case of 4 reeats, the events we are interested in are Success = ff F F F g Success = fsf F F; F SF F; F F SF; F F F Sg Success = fssf F; SF SF; SF F S; F F SS; F SF S; F SSF g 3 Success = fsssf; SSF S; SF SS; F SSSg 4 Success = fssssg The robabilities of the samle oints that make u each one of these events are all the same. and deend only on the number of success and failures. 4 (F F F F ) = ( ) 4 (SF F F ) = (F SF F ) = (F F SF ) = (F F F S) = ( ) 3 (SSF F ) = (SF SF ) = (SF F S) = (F F SS) = (F SF S) = (F SSF ) = ( ) (SSSF ) = (SSF S) = (SF SS) = (F SSS) = 3 ( ) (SSSS) = 4 ( ) Thus the robability model of a fail coin toss, the robability model is # Success (#) ( ) ( )4 4 ( ) ( )3 ( ) ( ) 4 ( )3 ( ) ( )4 ( ) 3 4 (#) If we move to n reeated tosses, we can redict the form of the result we will get # Success k = ; ; ; 3 : : : n n (#) np k () k ( ) n k where n P k is a natural number that counts the number of di erent ways k S s and (n k) F s can be lined u. Of course, the deends on the articular Bernoulli exeriment we are using, but because the samle saces deend only on lists of success or failure, the numbers n P k are the same for any Bernoulli trial. Here are some charts that illustrate the robability models for various and the fair coin toss where =.

12 n= =/ n=3 =/ n= =/ n= =/

13 n= =/ Again as the number of reeats gets larger, the robabilities of the extremes get so small, it ays to leave them o the chart.. n=3 =/ n= =/ Now all the scales in these charts are changing and they are di erent than the corresonding charts for the die toss. Still the growing similarities in these charts is striking. As the number of reeats of the Bernoulli Trial grows, the more the basic shae of the robabilities in the models look alike. In the die examle, it took a coule of trials to get started, but by the time we get to trials, the shae has begun to emerge. In the case of the coin i, the shae starts aearing quite early. In general, the closer the robability of success in the Bernoulli Trial is to, the faster the shae begins to aear. The further away it is from, the longer it take, but no matter how small, once the 3

14 number of reeats gets large enough, this shae will emerge in the chart of the robabilities..3 Binomial Coe cients The nal remaining mystery are the numbers n P k : Now, comuting one of these numbers requires us to count the number of "words" you can make using only the letters S and F. In articular, n P k give the number of n letter words that have exactly k of the letter S and (n k) of the letter F: Starting with n = ; the samle sace has only two one letter words fs; F g One of these has one S and the other has zero S s. That means P = P = For n = there are four words in the samle sace fss; SF; F S; F F g Grouing them by the number of S s gives So ffssg; fsf; F Sg; ff F gg P = P = P = For n = 3 there are four words in the samle sace fsss; SSF; SF S; SF F; F SS; F SF; F F S; F F F g Grouing them by the number of S s gives So ffsssg; fssf; SF S; F SSg; ff F S; F SF; SF F g; ff F F gg For For n = 4; it will be 3P = 3P = 3 3P = 3 3P 3 = ffssssg; fsssf; SSF S; SF SS; F SSSg; fssf F; SF SF; SF F S; F F SS; F SF S; F SSF g ff SSS; F SF F; F F SF; F F F Sg; ff F F F gg: And that leads to 4P = 4P = 4 4P = 4P 3 = 4 4P 4 = : 4

15 This is all leading to one of the most famous arrays of numbers in mathematics, Pascal s Triangle. This triangle looks like P P P P P 3P 3 P 3 P 3 P 4P 4 P 4 P 4 P 4 P P P P P P P P P P P P P P Each row stands for a number of trials n; the location in the row gives the number of S s we are allowed starting with. When we ll in the actual counts, an amazing attern emerges Every other row starts and ends with a. The rst row gets its two s, but we leave enough sace below and between them to write their sum. That sum is exactly the number that belongs in the second row. From then on, all the interior numbers in each row can be comutes by adding the two numbers in the revious row closest to them. In row, we see a followed by a, in the sace below and between them in row, we nd their sum. This is the third number from the left in row six, thus n = ; and (remembering to start counting the numbers in the row at zero) k = P = : This attern can be extended as long as you like. The numbers that aear are called The Binomial Coe cients: There is an alternate notation, and even a formula for these numbers: n n! np k = = k k!(n k)! where n! = n (n ) (n ) 3 k! = k (k ) (k ) 3 (n k)! = (n k) (n k ) (n k ) 3. These number come u so often in mathematics that most scienti c calculators have them built in. There may be a button or a command, but if a calculator does anything more than simle arithmetic, it usually also calculates the binomial coe cients somehow. Preared by: Daniel Madden and Alyssa Keri: May 9

1 Normal Distribution.

1 Normal Distribution. Normal Distribution.. Introduction A Bernoulli trial is simple random experiment that ends in success or failure. A Bernoulli trial can be used to make a new random experiment by repeating the Bernoulli