In this unit we investigate the link between experimental estimates of probability and theoretical probability and learn about short run variability. We take part in sports betting simulations designed to illustrate how damaging gambling can be.
 theoretically and experimentally examine the probabilities of games of chance
 describe the notion of short run variability
 estimate and find the relative frequencies of events
During this unit, students will explore issues related to gambling. In particular they will look at the chances of winning in the short term and the likely outcomes in the long run. As the teacher, you will need to be sensitive to the beliefs of parents before commencing this study. It is worthwhile to send home a short note explaining that the unit is about educating students about the realities of gambling so that as young adults they will be able to make informed choices. This unit is not an endorsement of gambling.
The unit explores how probabilities can be determined theoretically and experimentally and how these probabilities are used by organisations such as the TAB to determine what odds are offered. The unit should expand students’ understanding of probability. As we have said in the introduction, probability is an important concept that is used in a large variety of situations. An understanding of the notion and application of probability is essential in businesses such as insurance as well as in medical areas such as epidemiology. On the personal level, such knowledge enables us to understand the risks we take at the casino, the TAB, the stock market or with a medical operation. In some places the weather forecasters even assign a probability to the likelihood of rain falling the next day. Consequently students need to know the significance of probability and its implications in a range of settings.
Counters
Standard dice
Copies of the Multibet game board
Blank wooden cubes
Copies of the Horse Races game board
Paper bag
Playing cards
experimental estimates of probability, theoretical probability, variability, simulations, relative frequency, long run frequency, percentages, gambling, addiction, casino
Getting Started (12 sessions)
 Tell the students that they are going to simulate a game of netball played between New Zealand and Australia (Who else!). You have looked at the last few games played between the two teams to see how accurate the goal shoots were.
 Show them this table of results:
Team Goals Shots Taken Shooting Percentage Australia 248 293 New Zealand 223 329 Tell the students to discuss the results in their groups and to come up with some statements about the performance of the two teams in the last few games. You may wish to lead them with questions such as:
How many games between the two countries do you think are included in these results? Explain.
If the goal shoots of each team had ten shots, how many would you expect to be goals?
What is the purpose of working out the shooting percentage? 
Share the results of their discussions. Look for statements such as, The Australian shooters seem to get more in than the New Zealand shooters, The New Zealand shooters get more shots, The percentages are 84.7 for the Australians and 67.8 for the New Zealanders.
Ask why shooting percentages are calculated (percentages give a common base of one hundred so comparisons can be made). A key point here is that the shooters do not take the same number of shots so the common ratio is necessary to make fair comparisons.

To simulate the next game between New Zealand and Australia you are going to make two six faced dice. Each dice will have some faces labelled goal and some labelled miss. Challenge the students to create a dice for each country that closely matches the performance of the shooters over the past few games.
Give each small group two blank cubes to make their dice and justify their decisions. Black lettering could be used for the New Zealand dice and yellow/green for Australia.
The best solution is: New Zealand: four faces showing goal, two faces showing miss; Australia: five faces goal, one face miss. This solution is the best match to the shooting percentages in the initial table.
Students may use different reasoning to find their solutions. For instance, one way may be to argue this way. Two hundred and twentythree is roughly two thirds of three hundred and twentynine, and four is two thirds of six. So four of the faces of New Zealand’s dice will be goal and the other twowill be miss.
Another way might be as follows. Two hundred and ninetythree is about three hundred, which means that there are roughly fifty shots to a face. Two hundred and fortyeight is about five times fifty, so five of the six faces of Australia’s dice will be goal.

The next step in the session involves exploring the issues of short and long run frequency. Here we consider the difference between what happens with a small number of trials, against what happens with a large number of trials.
Tell the students that we are going to assume in the next netball test that the New Zealand shooters take the same number of shots as the Australian shooters. With their dice the students play a four shot game, that is each dice is rolled twice. The students decide who wins Australia or New Zealand. Draws are not recorded. The class results are then collacted as wins and losses. On a chart record how many wins this produces for Australia and how many for New Zealand. Extend this to a ten shot game (five of each dice) then a twenty and forty shot game, record the results on the chart as you go.
2 shots each 5 shots each 10 shots each 20 shots each wins losses wins losses wins losses wins losses NZ Australia  Ask the students to note any patterns they see in the table of data and why those patterns have occurred. Focus the discussion on the trend that the more shots taken, the more likely Australia will win. This shows that the greater the number of trials, the closer the results of the experiment will approximate to the longterm data.
Students should note that the results for a small number of shots are erratic and do not appear to match the theoretical chances closely. This is known as short run variability that is, it is difficult to predict the outcomes from a short number of trials.
Exploring (ideas for 35 sessions)
 Tell the students that they are going to investigate what happens in sports betting. Point out that it is important for them to realise that for some people gambling is an addiction that has disastrous effects on their lives. This investigation is designed to show how damaging gambling can be.
 Produce a section from the newspaper that gives the TAB odds for a sports event that will take place in the near future. For example, at one stage in the 1999 Rugby World Cup, Australia was paying $2.50 and the All Blacks were paying $1.85. This means that if a person bets $1.00 on Australia and they won the World Cup, then the person would be paid $2.50. On the other hand if Australia didn’t win, then the person would lose their $1.00. Ask: If someone had bet $10.00 on the All Blacks and the All Blacks had won, how much would that person have been paid for the win? (The answer is 10 x 1.85 = $18.50.)
 In this lesson we will simulate what happens when sports bets are placed. We use a netball test between New Zealand and Australia as an example. Tell the students that the dice they made yesterday will be used to find the outcome of the game.
Each student has $1.00 to invest. Tell them that the TAB is paying $1.40 for Australia to win and $2.80 for New Zealand to win. Ask: Why do you think the amounts being offered are different? (Past experience indicates that the Australians have a greater chance of winning  you can debate whether this is still likely!).  Get the students to write down which team they will invest in. If you wish, they can simulate going to the TAB and exchanging their one dollar coin for a chit. Collate the number of bets on each team and record this on the board, eg. Australia 20 students, New Zealand 10 students.
 Each group of students can then simulate the playing of the netball test using the dice from the previous lesson. They can either give each team a fixed number of shots, say thirty, or introduce another variable by putting the numbers 40 to 60 in a bag and drawing out the number of shots each team takes at random. Encourage the students to record the score in a tally chart.
Team
Goals
Misses
Australia
New Zealand
 Once the students have determined the result of the game tell the students to work out how much the TAB will pay out to winners for the game. Compare this with the total amount of money they collected ($1.00 for each student).
Discuss as a class in how many games the TAB got more money in than it paid out. Focus on which betting proportions would cause the TAB to make a profit or a loss. Given the quoted odds of $1.40 and $2.00 and a class of thirty students the outcomes for the TAB can be shown as:
TAB Profit or Break Even TAB loss Australia win Less than 22 people bet on Australia 22 or more people bet on Australia New Zealand win Less than 15 people bet on New Zealand 15 or more people bet on Australia Note that if the betting proportions fit in a particular zone the TAB will always make a profit.
People
betting on
Australia11 12 13 14 15 16 17 18 19 20 21 22 23 People
betting on
NZ19 18 17 16 15 14 13 12 11 10 9 8 7 The zone of TAB profit no matter who wins.
 Students may explore the scenario of setting the TAB odds for different games. It pays to start with simple situations first. For example, a penalty shootout in soccer between two teams where the dice used for both teams is goal, goal, goal, oops, oops, oops! In this scenario the students can explore what happens when different odds are set, like $1.50 for both teams.
 Summarise the exploration by asking the students what they have learned about sports betting. They may note that while some individuals win and some lose, the ultimate winner in the end is usually the TAB.
 Students can explore different ways in which odds are determined. These include:
Playing the Horse Race Game by rolling two normal dice. The sum of the dice gives that numbered horses a move of one space towards the finish.
Start 2 3 4 5 6 7 8 9 10 11 12 Finish After playing the game students will realise that some horses have a greater chance of winning than others due to the possible combinations of numbers that can be shown on the dice. The table below shows that horse seven has the greatest chance of winning. This is because 7 comes up 7 times out of the 36. This is more than any other number. (This is idea is also used in BEATIT.)
Betting on horse racing can be simulated by giving each student one dollar to place on their favourite horse. Point out before students place their bets that the more people who bet on a particular horse the less will be paid out for that horse.
Record the number of students who bet on each horse then show how the TAB calculates the how much is paid out for each winning horse.
From the total amount of money invested take away $10.00. For example, if thirty students invested $1.00, then the prize pool will be $30.00  $10.00 = $20.00.
The ten dollars is profit for the TAB. The pay out for each horse is then calculated by dividing the prize pool by the number of $1.00 bets placed on that horse. So if ten people have bet on horse five, it will pay $20.00 ÷ 10 = $2.00. Get students to calculate the other pay outs.
Simulate the race by playing the game and reflect on how many people win and how many lose. Again, in the long run the TAB is the only guaranteed winner!
 Another context for exploration is Lotto. Start simple by exploring the chance of picking two numbers out of three. The draw of Lotto can be simulated by putting playing cards, in this case A, 2, 3, into a paper bag, and having a person select two cards at random.
Note that the probability of winning can be worked out theoretically by finding the number of possible outcomes and what fraction of these are what you want. In threenumber Lotto there are three combinations, 1 and 2, 1 and 3, 2 and 3, since unlike Lotto Strike the order of drawing does not matter. The probability of winning is one out of three or 1/3.
Play several versions of Lotto, gradually increasing the complexity of the game. Discuss the proportion of students in the class who win each time and why this occurs. A possible sequence is: two numbers out of four (the chances of winning are 1/6); two out of five (the chances of winning are 1/10); three out of five (also 1/10); three out of six (chances of winning are 1/20).
In each case the probabilities can be found by making organised lists of all the possibilities. For example, in the Lotto game where two numbers are chosen from five, the ten possible outcomes are: 1 and 2, 1 and 3, 1 and 4, 1 and 5, 2 and 3, 2 and 4, 2 and 5, 3 and 4, 3 and 5, 4 and 5.
The key point here is that the more numbers there are to choose from, the less the probability of a win is.
In Lotto the prize pool is often set before the draw is made. In simulating this you can use a prize pool of twenty dollars ($30.00  $ 10.00 = $20.00 as before). In each game the pay out is found by dividing the prize pool by the number of winners. For example, if four students win then the pay out is $20.00 ÷ 4 = $5.00.
The pattern with prize money is that as the probability of winning decreases the potential prize money increases.
Reflecting
 The final lesson is based on a multiplication basic facts game called MultiBet, which is based on the casino game Roulette. You will need two dice labelled 4, 5, 6, 7, 8, 9, counters, and a game board for each group of players. In this scenario the students are placed in the shoes of Risky Betts, the Casino owner, who has to determine the pay outs for the game.
Introduce MultiBet to the class:
Each player starts with ten counters (their loot!).
They place bets in the following way:
The winning number is determined by tossing the two dice and multiplying the numbers that show (eg. 4 x 6 = 24).
If the winning number is not in those selected by a student then the casino takes all the counters.
If the winning number is one of those chosen by some students, then the casino must pay out but how much should they pay out?
As Risky Betts the students must decide what odds they will pay out on each type of bet remembering that the odds must be enticing to the players yet ensure in the long run that the casino makes money.
Once they have allocated odds such as 2:1, which means that $2.00 is paid out for every dollar placed, they can trial the game to see how their odds work in practice.
Note that there are twenty products on the board in total so that a bet covering four numbers has a four out of twenty (4/20 = 1/5) of being successful. The casino will want to offer odds of less than 5:1 if they are to make money in the long run.
2. As a means of assessing their progress in meeting the achievement objectives for this unit, ask the students to record the reasoning they used to decide how they allocated odds.
Some students may note that there are more ways for some numbers to occur than for others. For example, thirtysix can occur in three ways (4,9), (6,6), and (9,4), whereas fortynine can only occur in one way (7,7).
Dear Parents and Whanau,
This week in maths we have been exploring probability in interesting ways. Here is a problem to solve with your child.
Each time you buy a Happy Meal at McFunnels you get a card drawn form the Free Burger Box. The card will show the left, centre, or right side of a hamburger. When you have collected three cards that make a whole hamburger you can exchange it for the real thing!
Assuming that McFunnels put the same number of each card in their Free Burger Box to start with and each time you get a card it is purely by chance (random), how many Happy Meals will you need to buy to get a free hamburger?
Enjoy investigating this probability problem with your child. You might both be surprised.