Ideally, each group should have at least one TI-83 calculator. If there are not enough TI-83s in the class, a group with an 82 or an 85 will work. If there are still not enough calculators, some groups will have to use the random digits table. The first two parts of this activity give a brief introduction to generating random numbers on the TI-83 and TI-82 or 85 calculators. You only need to read these sections if you have such a calculator, and you only need to read the section about your calculator.

For all calculators, you should realize that the calculators essentially have a long list of random numbers in them. Most calculators will generate the same list of random numbers, each time you turn them on. You can reset the calculator to give different random numbers by changing the seed. On all of the TI calculators, this is done by storing a value to the variable named rand. Change the seed by selecting some HUGE number (of your choice) and storing the value in rand. To accomplish this, type out the huge number, press [STO>], then open the [MATH] menu and its [PROB] sub menu. Select rand and press [ENTER]. This is necessary to prevent all groups from generating the same data, so be sure to do this now.

TI-83 Random number generators

• rand Located on the [MATH][PROB] sub menu, this is used to change the seed (see above) and generate real numbers from 0 to 1. If you simply execute the rand command, it will return one number between 0 and 1. If you want 7 random numbers from 0 to 1, enter the command rand(7). This command is handy for uses similar to Table B, where you declare certain numbers to represent individuals with a property, and the remaining numbers to represent individuals without that property. The rand command gives decimals while the table has digits. Often, decimals are better. If you want samples of size 15 from a population where p = 0.47 have a property, assign values less than or equal to 0.47 as having the property, values larger than 0.47 as not having the property, and issue the command rand(15). Each of the 15 generated numbers which is less than or equal to 0.47 counts as someone who has the property, and those numbers larger than 0.47 indicate an individual without the property. The 15 numbers will appear in a single line. Use the arrow keys (left and right) to scroll through them all. Fortunately, this style of coding, or assigning values to decimals to indicate which do or do not have a property, is rarely needed for those of you with a TI-83. The commands below will, most often, remove the need for such assigning of values.
• randInt the 5th command on the [MATH][PROB] sub menu is used to generate any number of integers (whole numbers) from any starting value to any ending value (as in whole numbers from 1 to 24 or something). A single roll of a die produces a value from 1 to 6 (all values are equally likely). To simulate the rolling of 4 dice, just enter the command randInt(1,6,4) (see the left screen below). The screen indicates that you rolled three 3s and a 1. If you've labeled your population with numbers from 1 to 36, and you need a random sample of size 5, try the command randInt(1,36,5) (see the right screen below).  Interpret the numbers given as the random sample of size 5. The screen shows the sample 23, 12, 35, 27 and 22, so those individuals will comprise your random sample of size 5. If some value is repeated, issue the command randInt(1,36) (several times if needed) until a new person is generated.

The general syntax for this command is

This will give num integers, each with a value from start to end. Repeating the command (normally you can just type [ENTER] to repeat the last command) will give another sample of size num from the integers from start to end!

• randBin the 7th command on the [MATH][PROB] sub menu can be used for many simulations. Of particular interest for us is selecting a sample of size N from a population where p (a decimal like 0.05 for 5%, or a fraction like 1/6 for 1 in 6) represents the proportion of people who have a given property. If you simply want to know how many (out of N) have the property, issue the command randBin(N,p) and the calculator returns the total number of people who have the property. If you want a sample of size 100 from a population where 40% have a property, the command randBin(100, 0.4) will give the count (out of an SRS of size 100) which have this property. The left screen below indicates that 47 out of 100 have the property, so 53 out of 100 did not have the property.

On the other hand, there are situations where you need to know which individuals had the property and which did not. For example, if team A is better than team B, say team A wins 60% of all matches played against team B, and you are interested in knowing how long a best of seven series lasts (how many matches are required before one team wins 4 matches), you need to know more than who won the series. The command randBin(7,0.6) will indicate the number of games won by team A. If the result is 4 or more, team A won the series. If the result is 3 or less, team B won the series (so A lost). If the result is 5, you cannot tell how long the series lasted. Perhaps team A's record was WWWWWLL (5 wins then 2 loses), but it may also have been LLWWWWW or LWWWWL (or many other possible win-loss records with 5 wins and 2 losses). If you want to know not only who won the series, but who the winner of each match was, the randBin command can be used to tell you this as well. The command randBin(1,p,N) will return N values, each is a 0 or a 1. Interpret the value 0 as a loss and 1 as a win, in a series of games where wins occur with proportion p. The command randBin(1,0.6,7) will not only indicate if team A won the series (meaning there are 4 or more 1s returned), but which team won each game (a 1 means A won, a 0 means B won), so you can look at the result and decide how long the series lasted. The right screen above indicates that team A won the series in 5 games (team A won matches 1,2,4, 5, and 7, while team B won matches 3 and 6). Matches 6 and 7 never happened as the series was over by that time. When you need to know not only the total number in a sample who have a property (like A won a match), but which individuals possess the property and which do not, use the randBin(1,p,N) version of the command. Finally, I should indicate that you can simulate several samples of size N from a population with proportion p having a property, using the randBin command as well. Suppose you have a large supply of ball bearings and 10% (p = 0.1) are defective (too big, too small or not round enough). If you want 5 samples (lots) of size 200 from this collection of ball bearings, and only need to know the total number, in each lot of size 200, which are defective, the command randBin(200, 0.1, 5) will return the 5 counts. The screen below indicates that 20 in the first lot were defective, 27 in the second lot, 23, 21 and 24 in the 3rd, 4th and 5th lots were defective. The first number in the command is the lot size, the second is the proportion p and the third is the number of lots you'd like to draw.

TI-82 or 85 random number generators

• rand Located on the [MATH][PROB] sub menu, this is used to change the seed (see above) and generate real numbers from 0 to 1. If you simply execute the rand command, it will return one number between 0 and 1. This command is handy for uses similar to Table B, where you declare certain numbers to represent individuals with a property, and the remaining numbers to represent individuals without that property. If you want samples of size 15 from a population where p = 0.47 have a property, assign values less than or equal to 0.47 as having the property, values larger than 0.47 as not having the property, and issue the rand command 15 times (after the first time, simply press [ENTER] to re-issue the command). Each of the 15 generated numbers which is less than or equal to 0.47 counts as someone who has the property, and those numbers larger than 0.47 indicate an individual without the property. 
  
  The 15 numbers can all be generated at once using the sequence command. The command seq, located on the [OPS] sub menu of the [LISTS] menu, is used to make a sequence or list of numbers. The command seq(rand,X,1,15,1) will generate 15 random numbers between 0 and 1. On the TI-85, you can use x instead of X (just press the variable graphing key, [x,t,q,n] on either calculator). We do not need to know the entire syntax of the sequence command, but for those who are interested, the command seq(formula, var, start, end, step) will evaluate the expression formula (usually a formula with X in it) for each value of var (normally X) from X = start to X = end, with a stepwise of step. To square each of the even numbers, from 0 to 50, you could use seq(X^2,X,0,50,2).


If N is any whole number, the command N*rand will generate a number from 0 to N. This can make the assigning of values much easier. For example, a single roll of a die results in a 1, 2, 3, 4, 5 or 6. Each value (1 to 6) is equally likely. You can simulate the roll of a die with just the command rand, but then you will spend a great deal of time figuring out what 1/6, 2/6, 3/6, etc. are as decimals (numbers from 0 to 1/6 represent ones, numbers from 1/6 to 2/6 represent twos, numbers from 2/6 to 3/6 represent threes, etc.). Instead, the command 6*rand will generate numbers from 0 to 6. The generated numbers between 0 and 1 can correspond to rolling a one, numbers from 1 to 2 correspond to rolling a two, etc. Deciding if a number is between 1 and 2 is a lot easier than deciding if a number lies between 1/6 and 2/6. If you want to roll 3 dice, give the command seq(6*rand, X, 1, 3, 1). This command tells the calculator to generate a roll of a die for each value of X from 1 to 3 with steps of size 1. The screen below indicates that the first roll was a 5, the second a 3, and the third roll produced a 6. I reset my calculator to display fewer digits of these values. You can see the later values by scrolling with the left and right arrow keys. If you want to decide which team won each game of a seven game series, where team A wins 60% of matches against team B, use the command seq(rand,X,1,7,1) and interpret numbers less than or equal to 0.6 as wins for team A and numbers larger than 0.6 as wins wins for team B. The command asks for 7 values between 0 and 1 (the default of rand), since you run X from 1 to 7 with steps of size 1. The calculator will give you seven numbers and you'll need to scroll to see them all.

The assignment

1. Here is a description of a common game in establishments where small stakes gambling is allowed (or at least not patrolled real seriously by local police departments). For $1, you get to select a whole number from 1 to 6 (say you select a 5). The establishment's patron (bartender in common English) then gives you three normal dice (so all values from 1 to 6 appear equally often) and you roll all 3 at one time. The establishment's patron then pays you $1 for each die which happens to land on the number you selected. If you roll zero fives, you end up losing $1 (or you profit -1 dollars). If you roll 1 five, you get your dollar back and end up with a profit of zero dollars. If you roll 2 fives, you end up with a profit of $1 and if by luck you happen to roll 3 fives, your profit is $2. Let Y denote your profit from a typical play of this game. Y can take on the values -1, 0, 1, or 2. For this question you are to estimate the probability that Y = -1, 0, 1, and 2 and the average value of Y (which is your expected profit from this game). To this end, select a number from 1 to 6, then use the calculator or Table B to simulate 3 rolls of a die. Do this 5 times and record the information below. When you are done with these 5 plays of this game, record your values on the board to share your information with the class. Either wait for the entire class to put there data on the board, or go on to step 1 of the second question.

 

 

What value (1 to 6) did you select?              :

1. Each group should produce 5 simulated values of Y = your profit in this game. What is the total number of Y values for the class?

 

 

2. In the table below, record the frequencies (counts or total numbers) for the entire class's values of Y.

Value of Y	Class Count
-1
0
1
2

3. Compute the mean of this data set. Note: There are at least 2 shortcuts to doing this. If you enter the Y values in L1 and the counts in L2 and issue the command 1-VarStats L1, L2 (on the TI-83 anyway), the value of x-bar returned is the correct mean. You can also simply add the products of each Y value by its count, and divide by the total number of class observations. For example, if you have the counts 15, 8, 3 and 1 for Y = -1, 0, 1 and 2 respectively, the command (-1*15 + 0*8 + 1*3 + 2*1) / 27 will give the mean (27 is the total number of class observations, 15 + 8 + 3 + 1 = 27).
4. For each value of Y (-1, 0, 1, and 2) calculate an estimate of the probability that your profit in this game equals Y, by calculating the class count divided by the total number of Y values generated by the class. For example, if the class generated 55 Y values and 32 turned out to be zero, your estimate for the probability that Y = 0 is 32 / 55. Record your estimates in the table below.

Value of Y	Probability Estimate
-1
0
1
2

5. Let's abbreviate "the probability that Y = i", with the symbols p(i). The book claims that the mean value of Y is simply the sum of all the numbers i*p(i). In our case this is, -1*p(-1) + 0*p(0) + 1*p(1) + 2*p(2). Calculate this value from the table above. How does this "formula in the book" compare to your answer to part (c)?

 

 
 
 
 

6. Using your table above, calculate the probability that Y > 0 (meaning you win something), by simply adding the probabilities that Y =1 and Y = 2. Do you think this is a good game to play very often? Why or why not?

 

 
 
 

2. For this question we would like to estimate the average length of a best of 7 series between two teams. Team A is better than team B and team A wins 60% of all matches it has against team B. Best of 7 series between the 2 teams split naturally into two collections; those won by team A and those won by team B. We will estimate the probability that team A wins such a series, and two averages, the average number of games in a series where team A wins the series, and the average number of games in a series where team B wins. Using your calculator or Table B, simulate 10 best of seven series between team A and B. Note that the length of a series is the number of games it takes for one team to accumulate 4 wins. Record your series in the table below and then record your values on the board to share with the class.

1. Using all of the data from the class, what is your estimate of the proportion of series between these two teams which are won by team A? Explain how you estimated this value.

 

 
 
 
 

2. Using all of the data from the class, what is your estimate of the average length of a 7 game series between these two teams which is won by team A? Explain how you estimated this value.

 

 
 
 
 
 
 

3. Using all of the data from the class, what is your estimate of the average length of a 7 game series between these two teams which is won by team B? Explain how you estimated this value.

 

 
 
 
 
 

4. Why is it reasonable for the last average (length of series won by team B) to be larger than the second to last average (length of series won by team A)?