Heights of netballers and footballers

Transcription

1 CD40 SS Heights of netballers and footballers Heights of netballers Netball heights gives data showing the heights of all members of the Australian Netball Team, and the eight teams in the Commonwealth Bank Trophy (the AFL of netball). It is in the form of frequency tables. It gives the number of players of different heights for each team. 1 a One team is much higher than the others, on average. Which is it? b One team has the shortest player and the greatest range. Which team is that? c Find the mean height for each team. d The mean height for all players is cm. Are any teams well above or below the mean? 2 Find the median height for each team. (Notice that one team has 15 players, where the others only have 12.) Is the median close to the mean? 3 Find the height quartiles for each team. Over the same number line draw boxplots of the data. 4 Use the boxplots to compare the medians and spreads of the heights of all teams. 5 At the end of the regular rounds in the 1998 season, the eight teams had these points. Thunderbirds 23, Phoenix 23, Swifts 22, Ravens 16, Kestrels 13, Sandpipers 9, Firebirds 4, Orioles 2. Is there any relationship between the mean height of the team and its ladder position? Heights of footballers AFL heights gives data showing the heights of the members of all AFL team squads for At the bottom of the spreadsheet is the same data for all the players, taken together. It gives the number of players of different heights for each team. 6 The mean height of all players is cm. You can see a graph of the frequencies of all players heights on the spreadsheet. Describe the way they are distributed, including their range. 7 By looking at the tables, work out which teams have the highest and lowest average height. 8 Find the median height for each team. Compare this to the overall mean, which matches the overall median: cm. Are any teams well above or below the mean? 9 Find the height quartiles for each team. Over the same number line draw boxplots of the data. 10 Use the boxplots to compare the medians and spreads of the heights of all teams. 11 At the end of the regular rounds in the 1998 season, the teams had these points (in ladder order). Nth Melbourne 64, Western Bulldogs 60, Sydney 56, Melbourne 56, Adelaide 52, St Kilda 52, West Coast 48, Essendon 48, Richmond 48, Port Adelaide 38, Carlton 36, Geelong 36, Hawthorn 32, Collingwood 28, Fremantle 28, Brisbane 22. Is there any relationship between the mean height of the team and its position at the end of the season? Footballers heights compared to netballers heights There is a graph AFL and Netball on the AFL spreadsheet that has data from all players of both sports. This is given as a relative frequency, since there are far more AFL players than netball players in the top league. Relative frequencies allow comparisons to be made readily. They also allow probabilities to be obtained. 12 Use the graph to compare the heights of footballers and netballers.

2 CD41 SS Cumulative frequency graphs Graphs on paper Below are cumulative frequency graphs for all AFL and Austn Netball Association players for AFL players 1998 AFL players 1998 Cumulative frequency 120 Netballer heights Netball heights Cumulative frequency Cumulative frequency Cumulative frequency frequency Heights Heights (cm) 1 How many players are included in the data, for the AFL? for the Austn Netball Association? 2 What is the range of players heights, for both sports? 3 For each sport, find the height which has half the players under it, and half over it: the median. 4 For each sport, find the height which has a quarter of the players under it: the lower quartile. 5 For each sport, find the height which has a quarter of the players over it: the upper quartile. 6 Use these results to draw boxplots over the graphs above. Draw them at the level of the median. Spreadsheet 7 The graphs above are on the spreadsheets AFL heights and Netball heights. Using the data on the two spreadsheets you can create a similar graph for each team in the AFL and the ANA. Graphics calculator To change a frequency listing into a cumulative frequency listing on a graphics calculator you use a simple program which you can type in yourself. Firstly, put the heights into list 1. Then put the frequencies into list 2. Thirdly, make the first entry in list 3 equal to the first entry in list 2. Then return to the home screen. Then type: 0 X ENTER X+1 X:L2(X+1)+L3(X) L3(X+1) ENTER The first addition will appear on screen, and be added to list 3. Continue tapping ENTER until you get an error message. This means you have run out of data and the job is complete. Use STAT 1 to see that list 3 now has the cumulative frequencies. To draw a cumulative frequency graph, under STAT PLOT choose a line graph and use L1 and L3. Then ZOOM 9. 8 The data of the heights for AFL clubs is available to download from the CD for your graphics calculator. Do so for your favourite club and convert the data to cumulative frequencies. Then draw a cumulative frequency graph. 9 Answer questions 1 to 6 for your favourite club (cm) Heights (cm)

3 CD42 SS Distribution of Victorian incomes This page refers to the spreadsheet called Incomes. They are drawn using ABS data from the 1996 census which found out about the weekly incomes of all Victorians over the age of 14 years. They are divided by gender and by age group. The age groups are in age intervals of 10 years (25-34, etc) but with the lowest interval split into two parts (15-19 and 20-24). 1 Start by opening the graph All ages, all persons. a You will see that a significant fraction have either negative or zero income. Estimate the percentage of the total. b Note that the income ranges are not all the same width. At which incomes does the interval size change? c What is the modal range? Does this seem to be in the middle of the set? d Estimate the median income, the income which has 50% of people below and 50% above it. e The mean weekly income is about $393. This is about $90 above the median. Estimate the percentage of people whose income is below the mean. This shows that there are a number of people with very high incomes who pull the mean well above the median. 2 In the graph Income all ages we see the same data but divided by gender. a Which gender has more people with zero income? b Which gender has more people with incomes under $400 per week? (Look at all the lower income ranges.) c Which gender has more people with incomes of $400 or more per week? (Look at all the higher income ranges.) d What seems to be the median income for males? for females? (The calculated answers are: males $400, females $220.) e The mean weekly income for males is $437. This is well above the median. What does this show about the distribution of male incomes? f The mean weekly income for females is $301. Compare this to the mean male income, and to the median female income. Explain what each comparison means. 3 The age range years includes many students and unemployed people. Explain the shape of the graphs. Is there much difference between the graphs for males and females? Estimate the mean incomes for males and females in this age range. 4 The graph Income is drawn on the same scale as the others. Study it in comparison to the graph, and state the differences. Estimate the mean incomes for these males and females. 5 For the graphs Income 25-34, Income and Income there is a consistent pattern. a For which income ranges are there more females? b For each graph, estimate the mean incomes for males and females in this age range. c In which age range(s) is the difference between males and females the most obvious? d Describe the pattern that occurs in incomes as people grow older in Victoria. Do they get larger salaries on average? 6 A significant change to the shape of the graphs occurs at age What is it, and why? Estimate the means for the two gender groups. 7 Incomes for those over 65 tend to be limited to a small range. What is it? Why? (About 53% in this age range are females.) Estimate the means for the two gender groups. 8 The last graph is called Mean incomes by age and gender. This shows the means you have been asked to estimate, both males and females, for each age range. a Describe the shape of the two line graphs. b For two age ranges the average male earns in a week over $300 more than the average female. Which age ranges are these? c What percentage is the male income compared to the female income?

4 CD43 SS AFL and netball heights; traffic lights Height of footballers and netballers 1 The list show the heights (in cm) of the Essendon football team squad for , 189, 188, 176, 187, 192, 183, 179, 186, 186, 177, 187, 182, 200, 198, 183, 181, 193, 191, 178, 191, 185, 180, 186, 173, 195, 181, 182, 184, 188, 195, 182, 187, 190, 199, 194, 194, 184, 202, 199 Use these results to make a boxplot of the heights. 2 Use the results of question 1 to estimate the probability of randomly choosing Essendon footballers with these heights. a 186 cm b > 192 cm c < 180 cm d cm e 200 cm f < 175 cm 3 Use the heights in question 1 to complete the frequency table below. Height range (cm) Frequency Relative frequency b c Use the frequency table to draw a histogram. Add a row of relative frequencies to the frequency table. 4 Use the results of question 3 to estimate the probability of randomly choosing Essendon footballers with these heights. a cm b > 189 cm c < 180 cm d cm e < 190 cm f > 179 cm The table below gives the numbers of AFL and ANA players for 1998 whose heights were in the categories. Height range (cm) AFL frequency AFL relative frequency ANL frequency ANL relative frequency 5 a Use the frequency table to draw a histogram. b Add a row of relative frequencies to the frequency table. c Use the results of 5b to answer the parts of question 4 for the whole AFL. 6 Use the results of 5b to estimate the probability of randomly choosing ANA playesrs with these heights. a cm b > 179 cm c < 170 cm d cm e < 180 cm f > 169 cm The spreadsheets AFL heights and Netball heights has all this data. It also has the data for all the other 1998 teams in the AFL and ANA. Using the data, histograms have already been drawn using the intervals in questions 4 and 6. If you can, open the spreadsheet and view the graphs. 7 Estimate the probability that a randomly chosen player from each of the teams is in the height ranges given in question 4 (for AFL) and question 6 (ANA). Traffic lights When the traffic lights turn red you stop. What is the probability that you will be stopped by the lights? There are two ways to find out, and they may not give the same answer. 8 Find the time that the lights are red, and the total time of the light s cycle. The fraction of red time is one way of estimating the chance that you will be stopped. 9 Over a reasonably long period count the number of cars that pass the lights. Count also the number of cars that are stopped at the lights. The fraction of cars that stop is another way of estimating the chance that you will be stopped.

5 CD44 SS A TAB simulation The page refers to a spreadsheet program called TAB simulation. You will need to use this to tackle the investigations at the bottom of the page. Spreadsheet: TAB simulation 1 To make the spreadsheet work it is necessary to set the Iterations to 1. Choose Tools, then the Preferences menu, select Calculation and there you will find Iterations. Set it to 1. 2 On the spreadsheet you can change the amount of each bet. Set it at $20. You can change the starting balance. Set it at $100. You can change the TAB cut. Set it at 15%, by typing 15 and the % sign. At the start, cell B1 should read 0. 3 At the start there are 100 punters each with $100. You are one of them. You pick a number from 1 to 9, since there are 9 horses in the race, all equally likely to win. The other punters make their choices, randomly. 4 When you set cell B1 to 1 the first race takes place. The winner appears in cell B10, and all punters who choose the winner are highlighted. At the same time the bets are added, the TAB takes 15%, and the prize pool is the rest. For the first race TAB gets $300 and $1700 is left to be divided among those who bet on the winner. The winnings is the prize pool divided by the number of winners, often called the dividend. The winners all gain that much, and the losers all lose $20. (All values are rounded to the nearest dollar.) At this stage all 100 punters will still be betting. But later some will have no money left and will not bet. This is why the number of punters still betting is needed. The average amount held by all the punters is shown, and the highest value is highlighted. 5 There are three ways to make the next race happen on screen. change cell B1 (e.g. to 2) use F7 on Windows (command-= on Mac) change your pick. This will repeat the process. Note that the TAB gets another 15% (another $300) and the rest is shared among the winners again. Since the TAB has more, the average amount held by the punters will go down. 6 Continue for as many races as you like. Note that the TAB take goes up and the average held by the punters goes down. To start again, change cell B2 to 0. 7 If you wish to run a series of 8-race meetings quickly, you can change Iterations to 8. Then you will see eight races take place quickly whenever you change your pick or use F7 (cmd-=). Investigations How do these changes affect the rate at which punters lose their money? Explore the effects of: a changing the starting balance b increasing or decreasing the amount you bet each race. c changing the percentage the the TAB takes.

6 CD45 SS Cumulative probability Use the spreadsheet Probability Use command= to run 50 tosses. You can set the probability in cell E4. The program will randomly choose, and decide whether each result is a success or not. It will calculate and graph the relative frequency after each. This will be done four times, so four line graphs appear on the grid. The worksheet also displays the number of successes in each 50 trials. It uses them to estimate the probability (out of the 200 trials). You can compare this to the probability that you have set. 1 Run HEADS on your graphics calculator (or the spreadsheet Probability with p = 0.5) Do it until you have at least 20 trials. Sketch all the graphs roughly on the grid below. For each trial, record the final relative frequency on the dot graph. 2 Why is the first relative frequency either 0 or 1? 3 Why is there more variation in the first 10 or 20 tosses than later? 4 True or false? a Most final relative frequencies are between 0.4 and 0.6. b All final relative frequencies are between 0.35 and c If the relative frequency starts above 0.5 it usually stays above 0.5, and vice versa. 5 Repeat the investigation above with CUMPROB (or Probability ) using other probabilities.

7 CD46 SS Dice experiments There are two Excel spreadsheets. One is Dice and the other is How many rolls?. Die will let you set the number of faces on the die and roll it any number of times. It records the outcomes and finds all the relative frequencies. The graph shows these. For How many rolls?, the cell B1 must be blank before you start. When you change cell B1 to anything else, it will run a number of trials*. * The number of trials is set in the Worksheet template. Use Options: Calculation. Then choose Iteration and set the number of iterations as the number of trials you wish to run. Each trial is rolling a die (or whatever) until a success occurs, such as a 6. (It will be a die number if you set the probability to be.) The program will count the numbers of trials needed for each success, and graph the frequency of each number up to 20. It is quite possible that some number of trials are over 20. The numbers of possible trials are in column E and the numbers of times they occur are in column J. Using command = will give another set of trials. 1 Use Die (for 96 rolls of a 6-sided die) at least 10 times. Record the highest and lowest relative frequencies you get with 96 rolls. Note how close the relative frequencies are to. 2 If you use the spreadsheet Die you can vary the number of rolls, and observe how it affects the spread of the relative frequencies. Complete this table. 3 With Die you can also find the effect of changing the number of faces on the die. Repeat question 1 for 3- and 12-sided dice and compare results with the 6-sided die. 4 Use How many rolls? to find how many rolls you need to get one 6. Repeat 10 times and record the results here.. Find their average (mean): mean 5 The spreadsheet will let you do this any number of times, find the mean for you and graph the results. Try it and write a sentence describing what you find. 6 If you change the probability in the spreadsheet to something like 0.5 you will get a very different result. Will you get the chosen number (not 6!) in more or fewer rolls? Find out. 7 Find a relationship between the probability and the number of rolls needed to get the number.

8 CD47 SS Two-dice experiments The spreadsheet is called Dice. It rolls two dice as many times as you want, and graphs the relative frequencies for each total 2 to Run the spreadsheet Dice for 99 rolls. 2 Make a sketch of the graph you get on the grid. 3 Which number do you expect to be the most common? Why? Was it? If not, why not? 4 Run the spreadsheet Dice for 500 rolls. 5 Make a sketch of the graph you get. Be careful to show any differences from the last graph. 6 Are the results closer to what you expect? 7 Why or why not? 8 Run the spreadsheet Dice for 5000 rolls. Make a sketch of the graph lthis time. 9 Calculate the probabilities and enter the relative frequencies below. Use three decimal places. 10 Use all the results in the table above. Describe what happens as you use more rolls.

9 CD48 SS Coin tossing experiments There are two Excel spreadsheets. They are Coin tosses, and Winning Streaks. Coin tosses lets you toss any number of coins (up to 10) as many times as you wish. It records the result of each toss and adds the numbers of heads. It records these in the frequency table and plots a graph of the results. Winning streaks is concerned with runs of wins, such as in a football season. However it can just as well be applied to runs of Heads with coins. - Start by setting cell B2 to 0. This resets all the counters to 0. - For a coin tossing simulation, set the probability to Then set B2 to the number of trials you want. (It is valuable to set it to 1, then 2, then 3 and so on, until the winning streak is over, and see how the template and the graph change.) - Winning streaks will display the result of a game (or coin toss) as either W or L. If it is W, it goes into the winning run box. This continues until the winning streak is over. The run length is displayed and is recorded in a frequency table (using cells B to U in row 8) that becomes the basis for the graph. - Winning streaks also records the number of games played, calculates the percentage of wins (so you can compare this to the probability) and the average run length. 1 Run Coin tosses for one coin only. Do this 20 times. For each trial, write the greatest number of either heads or tails here. 2 Use Coin tosses to toss 1 coin for these numbers of tosses. Write the relative frequencies of heads or tails. What happens as you use more tosses? 3 Run Winning Streaks (with p = 0.5) and write the number of each run length. 4 Find a mean. (Winning streaks does this for you.) 5 Repeat questions 3 and 4 several times to feel sure of what the average run length is. 6 Use Winning streaks to repeat the investigation, but change the probability. Find a relationship between the average run length and the probability.

10 CD49 SS Card experiments The spreadsheet Cards simulates choosing at random a standard pack of 52 cards. Set the number of cards you want to select in cell B1. You can find the number of cards: of each colour (red or Black) of each card face (1, 2, 3,,10, J, Q, K) of each suit (hearts, diamonds, spades, clubs) The spreadsheet records the number and find the relative frequency for each suit, reds, blacks, number cards, face cards and aces. It also records the number and relative frequencies of many compound events. 1 Perform 100 choices. 2 Enter the relative frequencies into the table below. 3 Calculate the probabilities and write them in the table below. 4 Compare the relative frequencies with the probabilities. 5 Multiplication law (overlapping outcomes) Compare these values. Pr(black AND face) with Pr(black) Pr(face) = = Pr(red AND number) with Pr(red) Pr(number) = = Pr(club AND face) with Pr(club) Pr(face) = = 6 Addition law (non-overlapping outcomes) Compare these values. Pr(black OR heart) with Pr(black) + Pr(heart) = + = Pr(ace OR face) with Pr(ace) + Pr(face) = + = 7 Addition law (overlapping outcomes) Compare these values. Pr(black OR face) with Pr(black) + Pr(face) Pr(black) Pr(face) = + = Pr(red OR number) with Pr(red) + Pr(number) Pr(red) Pr(number) = + = Pr(club OR face) with Pr(club) + Pr(face) Pr(club) Pr(face) = + =

11 CD50 SS Several coins at once The spreadsheet for use with this page is called Coin tosses. Start with a 0 in cell B3. This resets all the counters to 0. Set the number of coins, maximum 10. Now set B3 to the number of trials (tosses of that many coins). The spreadsheet will run and show the results of the tosses for each trial. It counts the number of heads and enters them into a frequency table, in cells D3 to E13. This table is the basis for the graph. You may interrupt the program while it is running by clicking on a cell in the worksheet. It will restart after a short time. 1 Run Coin tosses for three coins (90 trials). Do it 20 times. 2 Sketch a graph of the relative frequencies on the grid. 3 Why are 1 and 2 heads more likely than 0 or 3 heads? 4 Draw a tree diagram and compute the theoretical probability of each of 0, 1, 2 and 3 heads. Pr(0 heads) = Pr(1 head) = Pr(2 heads) = Pr(3 heads) = 5 Compare the theoretical values to your relative frequencies. 6 Run Coin tosses for 4 coins and 90 trials. 7 Sketch a typical graph. 8 Draw a tree diagram and compute the theoretical probability of each (0 to 3 heads). Pr(0 heads) = Pr(1 head) = Pr(2 heads) = Pr(3 heads) = Pr(4 heads) = 9 Compare the theoretical values to your relative frequencies. 10 Explain why we should expect these graphs to be symmetrical. 11 Investigate the shape of the graph for 10 coins

12 CD51 SS Chuck-a-luck: a three-dice game You are given a bank of $10. Each game costs $1 to play. You select a number to play a game. Three dice are rolled and their numbers are displayed on the screen. If your number appears once, you win $1, and get your $1 entry back. If your number appears twice, you win $2, and get your $1 entry back. If your number appears three times, you win $3, and get your $1 entry back There is a spreadsheet for this page, too. It is called Chuck-a luck. Start by setting cell B2 to 0. This resets all the counters. Then set it to 1. It will run one game, so that you can see how it works. It displays three die numbers. The number chosen is 6. (You can change this in cell H2. You must Unprotect the document. There is no password.) It records the number of triple 6s, double 6s and single 6s in the table, and multiplies this by the prize money to record your winnings. Whenever you win it returns the $1 entry as well. It gets the total money paid out, and the total taken in, and finds the difference (the win or loss). To get an idea of whether the game is profitable, it finds the win or loss per game. Play a large number of games to see how fair it is in the long run. (You may change the prize money by Unprotecting the document first.) 1 Play the game one roll at a time to see how it works. 2 Run Chuck-a-luck 20 times. Keep your own tally of how much you have in the bank in the table. 3 On the spreadsheet Chuck-a-luck, you can change the prizes for matching one, two or three dice. It will do all the recording and calculating for you. Find prizes that a are entirely fair no win or loss in the long run b make you lose 50 cents per game! c let you win 50 cents per game! 4 When you roll three dice, how many possible triples are there? So what is the probability of rolling three of your number (say 6) at once?

13 CD52 SS The chance of a double 6 or a matching birthday There are two Excel templates for this sheet. One is called Coincidences. For Coincidences, start with cell A1 set to 0. In cell A2 set the probability. For a double 6 you need 1 chance in 36, so enter =1/36. You may try as many rolls per trial as you like up to 40, rolling two dice. Now set the number of trials (A1) to the number you want. (It is valuable to set it to 1, then 2, then 3 and so on, and see how the template changes.) The results are shown in the boxes on the right. When there is a successful roll, one or more cells show a 1. This makes a successful trial. The successful trials are counted and the percentage of trials that are successful is displayed. The other spreadsheet is Birthday. We are trying to find the chance that, in a number of people, two share the same birthday. Start by setting cell D2 to 0. Set the number of trials you want and the size of the group; then set D2 to 1. The spreadsheet will run the trials, and determine when each match occurs. It will count the successful trials, and work out this as a percentage of the total trials. The table and the graph, which is based on the table, give the theoretical results. You can compare the percentage with the theoretically expected outcome. 1 Guess: What is the chance that when 20 people roll two dice at least one rolls a double 6? 2 Use DOUBLE6 or Coincidences to roll two dice 20 times. Each time, note whether there is at least one double 6 (write a 1) or no double 6 (0). 3 What percentage of trials were successes? Compare this to your guess. 4 With Coincidences, increase the number of trials of the same experiment (20 rolls) to 100 or even 1000, and run. This will give you a better idea of what happens in the long run. Write about it. 5 With either DOUBLE6 or Coincidences, change the number of people involved and find the relative frequency. Estimate the probability and then record the relative frequency for at least 10 trials. 6 What happens in question 5? Explain it. 7 The Birthday problem. What is the chance that, in a group of people, two share the same birthday? Estimate the probability for different numbers of people. Then test. 8 What happens in question 7? Explain it.

14 CD53 SS Mini-Lotto, or 6 from 45 The spreadsheet for this page is Lotto. Start by setting cell B3 to 0. This will reset all the counters. Choose the number of numbers. For MiniLotto, choose 6. For the real thing choose 45. Choose how many numbers are to be chosen. For MiniLotto choose 2. For the real thing choose 6. To play one game, ask for just 1 trial. It will show the results. To play many games, ask for more trials. The spreadsheet will flash the results, and the graph will show a column graph of a frequency table at the end. This will show which numbers are most likely to be chosen. (The start of the frequency table is visible at the bottom of the screen.) It shows the number of winners (all numbers) and the percentage of winning trials. 1 Use 6 numbers and choose 2. Lotto selects your numbers for you. Do this and run one trial. Observer whether it shows 0, 1 or 2 correct guesses. (Lotto only shows first division winners.) 2 Run at least 100 trials. 3 Find the relative frequency for guessing correctly both numbers. 4 Work out the theoretical probability for this by listing all the possible pairs that can be guessed. The winner is one of these. 5 Compare the theoretical probability with your relative frequency. 6 With the spreadsheet, use many more trials to see if the relative frequency gets closer to the probability in the long run. 7 With Lotto you can vary the game and see how the relative frequency of success changes. 8 Lotto also graphs the frequency of the numbers in the game. You can see whether they really are equally likely or whether the game is biased. 9 In real 6-from-45 lotto, is the set 1, 2, 3, 4, 5, 6 more or less likely to win than any other set of six numbers? Explain.

15 CD54 SS AIDS testing and Smith family There is are two spreadsheets for this page; one is called AIDS testing and the other Smith family. AIDS testing AIDS testing has no graph. Start by setting B1 to 0. This resets all the counters to 0. Set the probability of HIV positive to a low value. Set the probability of a true test to a high value. Set the number of tests to 1. Set cell B1 to 1. The result is shown at the top right of your screen. These results are added to the two-way table in the centre of the sheet. The spreadsheet counts the number of negative tests (where the result says that the person is not HIVpositive) and also the positive tests. The bottom right corner gives two percentages: the percentage of the people who test positive who actually are HIV positive. the percentage of the people who test negative who actually are HIV positive the false negatives. Increase the number of tests gradually to see how the spreadsheet works. Then try a large number of tests to see how the percentages work out. 1 Run a trial of one test, to see how it works. Continue this until you see all the possible cases. 2 Those that are told they are HIV positive are not all HIV positive. The percentage of false positives is given on the spreadsheet. Complete the table by changing the probabilities to the values given. 3 False positives should be limited if possible. Explain how they arise, and what values of the probabilities give rise to greater numbers of them. Smith family The information is that one of the two Smith children is a girl. The problem is: What is the probability that the other child is also a girl? Start with cell C2 set to 0. This resets the counters. Cells A2 and B2 will show the gender of the two children. Cell A4 will show 1 when either child is a girl. These are tallied in A6. Cell B4 will show 1 only when both children are girls. These are tallied in B6. Cell C6 shows the percentage of cases for which both are girls, given that one is a girl. Run a small number of trials to see how it works. Run a large number of trials for the simulation. 4 Most people think the probability that the other child is a girl is 0.5, since it could be either a boy or a girl. What result does the spreadsheet give? Which do you believe? 5 Explain why the answer is not 0.5.

16 CD55 SS Four marbles and two cards Four marbles The spreadsheet Four marbles simulates an activity in which two black and two white marbles are rattled around randomly in the base of a bowl. They will eventually come to rest in one of two positions: with the colours opposite each other with the colours side-by-side. What is the probability that they will be opposites? Set cell F6 as 0 to reset the counters. Then set a number of trials. (If you start with 1, then change to 2, etc., you will see clearly how it works.) For the simulation, you need a large number of trials. 1 Estimate the probability that the marbles will be opposites : 2 Try it. Write your relative frequency here. 3 Explain what happens and why the probability is as it is. Two cards The spreadsheet Two cards simulates an activity in which two identical cards are used, white on both sides. However, one is marked A on both sides and one A on one side and B on the back. You choose one of the A sides at random. and place it on the table, face up. Another person then has to guess whether the back is A or B. Here are the questions. Guess the answers now. Which is most likely, or are they the same? What is the probability that the back is also A? Start with cell B8 as 0. This will reset the counters. Then set a number of trials. (If you start with 1, then change to 2, etc., you will see clearly how it works.) One of the A sides is chosen at random. The letter underneath is displayed. A count is kept of the hidden As and hidden Bs. For the simulation, you need a large number of trials. 4 Estimate the probability that, if you can see an A, the back of that card is B. 5 Try it. Write your relative frequency here. 6 Explain what happens and why the probability is as it is. 7 Use a tree diagram or list to show all the possibilities. Select only those that have an A as the first choice. From the A-first cases, how many have B as the second choice? What is the probability of B-second, given A first?

17 CD56 SS The quiz show puzzle You have been presented with three identical cups. There is a coin under one of them. You choose one of the cups. The probability of guessing correctly is 1 in 3. Now you are shown that the coin is not under a different one of the cups. (The person who put the coin under one cup knows which it is.) You are offered a chance to change your mind, to switch to the other cup not chosen by you, and not revealed by the other person. The problem is: Should you change your mind? Are you more likely to win if you change, if you do not change, or does it make no difference? You should trry to work out what you think is the situation now. Write your ideas here. The spreadsheet Quiz show allows you to explore this problem. It imitates every step above. In cell A4, it chooses a cup at random under which to place the prize and displays its choice. In cell A6 it randomly chooses for you and displays your choice. There is now either one or two cups which do not have the prize and are not your choice. The spreadsheet selects one (if there are two) or chooses the only one left. This is the EMPTY cup. Now either the guess changes or not. The computer will randomly do this for you. If it changes and there is a choice, the computer will choose one or the other, randomly. The computer will display whether you win as a result of changing or not changing. It keeps a count of No Change wins, and New choice wins, and calculates the percentages. 1 Run 1 trial (one contestant) to see what happens. Continue doing this until you see how the computer matches the description above. It is important to understand how it works. 2 The spreadsheet picks a coin cup randomly. Sometimes it will match you changing your mind, and sometimes it will match you not changing. Run it many times and see in which way you are more likely to win: by changing or not changing. 3 Now, can you explain what happens?

18 CD57 SS The game of Beetle The game Beetle requires you to roll a die to obtain body parts and construct a beetle. You must get a body before you can add the head or the six legs. You must get a head before you can add the two eyes, two feelers or mouth. The minimum number of throws needed to get a beetle is 13, the total of the last row in the table. To achieve this is MOST unlikely. Play the game with a die, and see how long it takes to get the whole Beetle. The spreadsheet Beetle plays the game just as you would. Set cell B1 to 0. This resets all the counters. Then try just one game slowly to see what happens; slow it down to see the effect of each roll separately. Here is how. Use Options, Calculation, and turn Iteration to 1. Then use Command = (or F9) to roll a new die number each time. Cell D2 will display the number on each die roll. The body will appear only on rolling the first 6, and so on. It finds the total number of rolls needed to complete the game. Run many trials at high speed, by resetting Iterations to The spreadsheet notes the smallest score for all the games played and finds an average. Each score is added to a frequency table, which forms the basis for the histogram graph. 1 Play one game. Do this several times until you see how it works. 2 What is the probability of getting Beetle in less that 20 rolls? Guess: Try it: 3 Run the game many times and complete the table. You will need to add some relative frequencies. 4 What seems to be a likely average score for Beetle? 5 Sketch a typical histogram of the scores. 6 How many rolls do we expect to take to get a 6 and then six 1s? 7 How likely is it that the legs are the last parts to be rolled?

19 CD58 SS Betting: equal and unequal odds The Excel template Equal odds assumes that the horses are equally likely to win. Start by setting cell H1 to 0. This will reset all the counters. You will see that you start with a balance of $100. You bet on horse 1 to win each race. Set the odds in cell H3. The spreadsheet allows you to change these odds. The horses are die numbers, so fair odds are 5 : 1. Set a bet strategy in H5. If you type a whole number, you will bet that number of dollars each time. If you type r, you will bet a randomly chosen number of dollars each time. If you type a decimal under 1, you will bet that fraction of your balance each time. (For example 0.5 means that you bet half your previous balance on each race.) Start with a run for one punter. Use a $10 bet each time for the first try. You will see that your bet is always $10. The winning number is in the column headed w. When horse 1 is not the winner, you lose your bet. Your balance goes down. When horse 1 is the winner, you win your bet. You gain five times the $10, i.e. $50. The spreadsheet shows the balance after eight races. The end balance is added to a frequency table, which is used for the graph Equal chances end balances. This gives you a picture of how much the punters had at the end of the eight races. It also gives the percentage of punters who did not lose at the end of the day. (They still had $100 or more.) 1 Start with odds of 5 : 1 and bet $10 per race. Try one punter, for at least 10 race meetings. Write down the end balance for each race meeting. What seems to be the chance of winning (that is, having $100 or more at the end)? 2 Now run the race meeting for many punters and check the percentage of winners. 3 The odds 5 : 1 are fair. Change them to 4 : 1. These are not fair, as $1 in every $5 stays with the bookmakers. Now what seems to be the probability of winning? 4 Complete the table. How do the odds affect your chances? The spreadsheet Unequal odds is very similar. However it is more realistic. This time the odds for each horse can be different. The spreadsheet converts the odds to probabilities and adds them. This total probability should be about 1. If bookmakers are involved the probabilities will add to about 1.2, to allow them to make some small profit in the long run! The probability of each horse (1 to 6) winning is based on the odds. Bet in the same way. 5 Repeat the questions 1 to 4 above for your choice of probabilities. 6 Does the favourite (the horse with the shortest odds closest to 1 : 1) always win? 7 How does the total probability affect the percentage of winners?

20 CD59 SS Craps Craps involves two dice, which you roll together on your turn. If you roll a total of 2, 3 or 12 (craps) you lose your turn and win nothing. If you roll a 7 or 11 (a natural) you win and have another turn. If you roll any other number (a point) you keep rolling. If your score is repeated before a 7 appears, you win. If a 7 comes first you lose your turn. The spreadsheet Craps is designed to let you enter a bank balance and bets. After each set of rolls it keeps tells you your balance. It keeps the same amount bet until you have gone broke, or decide to quit. Enter 0 in cell D1 to start, then a number for the number of games. Enter a bank size and a bet size. The spreadsheet will keep note of your balance as you play many games. The first roll is in cell A1. For 2, 3 or 12, cell B2 says CRAPS and cell B3 says LOSE. For either 7 or 11, cell B2 says NATURAL and cell B3 says WIN. You get another turn. For other numbers (4, 5, 6, 8, 9, 10), cell B2 says POINT, and more rolls appear in column 1. If a 7 appears before the point, it will be LOSE in cell B3. If a repeat of the point appears before a 7, it will be WIN in cell B3. The spreadsheet will graph the balances of the first 30 games. 1 Run it once to see how it works. 2 Run 100 trials and keep your own record of the outcomes. Use this tally table. (Use ). 3 What seems to be the probability of a win at Craps? 4 Now calculate the probability of a win and see how it compares with the spreadsheet. a What is the probability of 2, 3 or 12 (Craps)? b What is the probability of 7 or 11 (Naturals)? c What is the probability of 4, 5, 6, 8, 9 or 10 (a Point)? 5 The probability of winning after a point is about 0.4. So what is the probability of getting a point and then winning? 6 Put it together. What is the probability of a win at Craps? How does this compare with the experiment?

21 CD60 SS Radioactive decay In this simulation you have 100 atoms, simulated by dice. At each throw of 100 dice those showing a certain number are removed, having decayed. The spreadsheet is called Decay. It displays 1000 atoms, with 1 in each cell. Start by setting cell K13 to 0. This will reset the counters. Enter a 6 in cell K13. This will simulate rolling dice, so that the probability that an atom will decay is 1 in 6. The spreadsheet will remove some of the 1s at each turn and show the number left after each throw. This will continue until all have been eliminated. The graph will display a line showing the number left after each throw for the first 50 throws. The graph also calculates the theoretical half-life the number of throws at which we expect half of the 1000 to be left. You can compare the simulation with the theory. At the end, reset with 0 in cell K13. For any other probability, use a different number in cell K13. 1 Run Decay with 6 faces on the die until all the 1000 atoms have gone. Enter the number of dice left after each roll in the table below. 2 Sketch a graph of these values by copying the graph provided by the computer. 3 Use the results or the graph to estimate the number of throws needed to get from: 1000 to to to to 250 This number, needed to reduce whatever number of dice to half that number, is called the HALF-LIFE. 4 Change the number of faces on the die to 4. Run it. Describe how it is different from the 6-faces result. 5 Now change the number of faces on the die to 10. Run it. Describe how it is different from the 6-faces result. 6 Complete this table.

22 CD61 SS Lines of fit and predictions On this page are a number of scatterplots. a Describe the association (strong, weak or none), positive or negative. b For those with some association, draw an ellipse that just includes most or all of the points. c Use the ellipse to help you draw a line of fit. d Use the line of fit to help you determine a likely value for y when x = 1.5 and 7. e Use the ellipse to help you decide on the range of possible values that y could have. 1 (1.5, ± ), (7, ± ) 2 (1.5, ± ), (7, ± ) 3 (1.5, ± ), (7, ± ) y 6 y 6 y x 0 x 0 x (1.5, ± ), (7, ± ) 5 (1.5, ± ), (7, ± ) 6 (1.5, ± ), (7, ± ) y y y x 0 x 0 x (1.5, ± ), (7, ± ) 8 (1.5, ± ), (7, ± ) 9 (1.5, ± ), (7, ± ) y y y x 0 x 0 x Explore this further using the spreadsheet Sampling lines of fit.