Office and Administration  BOOKMAKER
Probability:
Horse Racing Game (from
TES)
Primary
Middle
Australian
Curriculum General Capability:
Numeracy
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.
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 (PDF)
In this lesson we conclude the unit with a game that involves probability.
Introduce the Horse Race game (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
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 MultiBet. 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.
1. 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 (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, thirtysix can occur in three ways (4,9),
(6,6), and (9,4), whereas fortynine 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).
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 5050 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 6horse race
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 FavouriteLongshot
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?
