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Probability: Horse Racing Game (from TES - Times Educational Supplement)

Primary Middle

Australian Curriculum General Capability: Numeracy

Horse Game

Teacher

A good little game to get students thinking about sample spaces and probability. Students choose a horse to bet on, numbered between 1 and 12. Two dice are then rolled and the horse with that total moves forward. Students intially will bet on their favourite numbers, leading to a good conversation about how we can tell which numbers will be more likely to win.

"Word of warning: the first time I used this number 11 won, which was less than ideal for illustrating the probabilities invovled!"

Here is the enlarged image. Print two copies - one for each half of the class. You will need to laminate to keep the "racecourse" looking good.

Horse Game

Probability: the Horse Race Game (from NZ Maths: What are the Chances - Session Five)

Middle Secondary

Australian Curriculum General Capability: Numeracy

Australian Curriculum General Capability: Critical and creative thinking

Australian Curriculum General Capability: Personal and social capability

Australian Curriculum General Capability: Literacy

Teacher: Lesson Plan [below]:

1. Play the Horse Race Game (Local Copy: PDF)

In this lesson we conclude the unit with a game that involves probability.

Introduce the Horse Race game (Local Copy: Copymaster Four) and have three students play a demonstration game for the class. The gameboard explains the rules. you will need the gameboard, two standard dice, and two counters of one colour each (their racing colour).

2.
Put the class into groups of three and have them play five games each.
While students are playing, roam the room. Look for the following:
Do students notice that the distance horses need to gallop is different.? For example, Horse One must travel 11 steps but horse Five only needs to travel three steps.
Does that seem fair?
Do students notice that differences of zero, one and two occur frequently but differences of three, four and five are less frequent?
Do they notice that a difference of six is impossible?

3. After five games for each trio gather the class to discuss their observations.
Which horse has the best chance of winning? Why?

Students should comment that the horses that have the furthest to go move more often and the horses with the least distance to go move least often.
Why do some horses move more often than others?
How could we find all the possible outcomes, when two dice are rolled, and we find the difference?

4. Invite students to work out the sample space with a partner, using whatever method they like.

5. Allow students some time to develop a theoretical model. Gather the class to share their thoughts.
Is 1 on the first dice and 5 on the second dice the same outcome as 5 on the first dice and 1 on the second dice?

How many different outcomes are there? (36 possible outcomes)

5. Show Slides One and Two of Local Copy: PPT. The tree diagram shows how all 36 possible outcomes occur.
How many outcomes give the event of Horse Zero moving? (Six, (1,1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6).

6. The table shows the differences produced from the set of outcomes.
What patterns can you see in the table? (Students might note the diagonal arrangement of cells with the same differences)
Which horse has the best chance of moving? How do you know?
Can you find the probability of Horse One moving on a single throw? (10/36 ≈ 28%)

7. Discuss the probability of other horses moving on a single throw.
Is the game fair? Does each horse have an equal chance of winning the race?

8. Students might match up the probability of each horse moving on a single throw and the number of steps the horse needs to win.

Horse	0	1	2	3	4	5	6
Steps to win	6	11	9	7	5	3	2
Probability of moving on one throw	6/36	10/36	8/36	6/36	4/36	2/36	0/36

The table shows that the distances balance the probabilities well except for Horse Six that has no chance of moving.

9. Develop a Horse Race game in which the dice numbers are either added or multiplied. Make the game as fair as possible for the horses.

Assessment

Evaluate students’ understanding of probability using the multiplication basic facts game called Multi-Bet. You will need two dice labelled 4, 5, 6, 7, 8, 9, counters, and a Local Copy: 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.

1. Introduce Multi-Bet 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 (e.g. 4 x 6 = 24).
If the winning number is not in those selected by a player, 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. How much should the Casino pay out for each type of bet?

The odds must be enticing to the players yet ensure in the long run that the casino makes a profit.

2. Once they have allocated odds such as 2:1, which means that $2.00 is paid out for every dollar placed, students 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) chance 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.

3. 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, thirty-six can occur in three ways (4,9), (6,6), and (9,4), whereas forty-nine can only occur in one way (7,7).

4. Has the board been designed to separate numbers that have a higher chance of occurring? Could you design the board better?

5. How do the varying probabilities affect the odds that the Casino should paid out? (from NZ Maths: What are the Chances - Session Five)

Your Best Friend at the Races - Maths! Know when to bet, and when to hold (from University of Melbourne)

Secondary

Australian Curriculum General Capability: Numeracy

Information (from University of Melbourne)

The first step is to find how likely the bookmakers think a horse is to win a race (the probability).

Bookmaker's Probability

For example, if a horse has odds of $2, then the bookmaker’s probability is .5 or 50%

If you’ve had a look at the form guide or heard an inside tip, you might feel that a horse is really more likely to win than the bookmakers suggest. If so, it’s time to bet.

Why should we bet now? If you’re correct in saying that a horse is more likely to win than the bookmakers suggest, you should expect to make a profit ‘on average’ from bets on these horses. Now the first few times you place a bet you may not have a win. However, assuming you’re correct that the bookmaker underrates the horses you are betting on, over time you should make a profit.

To explain this, consider tossing a coin. In the long run we expect 50% of the results to be heads, and 50% to be tails. But if you toss a coin only a few times you may not get close to 50%. However as you toss the coin more and more times, the results should get closer and closer to a 50-50 spread of heads and tails. This is known as the law of large numbers. Hence the more races we bet on that we expect to make a profit from… the closer our budget will approach a profit (the profit we expect given the probabilities).

The $pecial equation

Now we know when it becomes profitable to bet, but how much should we bet? Our entire budget? A quarter?

This equation, which forms the basis of the Kelly betting system, suggests an amount to bet on a horse given the odds, the probability you think it has of winning and your total budget. It can be mathematically proved that this system, assuming you have accurate probabilities, produces maximum income.

Example: A 6-horse race

Table

Risk and Reward

Betting the amount given by the equation will maximise your income if your probabilities are sound. But what if you’re not confident? Well, a common way to reduce the risk of each bet is to bet half or a quarter of the amount suggested by the equation. Whilst this does reduce potential income, the risk is greatly reduced, and if your probabilities are inaccurate it will not strip as much money away from you relative to the full bet.

The main drawback of this betting system is that it relies on having a belief of how likely your horse is to win a given race. Of course the ordinary punter probably doesn’t have the knowledge to put an exact number on how likely a horse is to win. However services do exist which suggest probabilities of winning races. Complex algorithms, the foundation of such predictors, take into account a range of past data which can be quite accurate.

Another interesting point to take into account is the Favourite-Longshot bias, where both individuals and markets tend to overvalue the chances of underdogs and undervalue the chances of favourites.

So there you have it, statistics can help give you the best chance to leave the track with a little bit more weight in your back pocket.

1. As a pair, obtain the racing form from the newspaper. Look at one race. List the same content as the table above.

2. Just for fun and hypothetically, say your total budget is $53. Work out whether it is profitable to bet or not.

3. Look at the results either online or in the paper. How did you go?