Abstract. Probability is an important topic in the high school mathematics
curriculum. However, dealing with randomness is always a challenge for
Vietnamese students because they have not been encouraged to use their intuitive
abilities and they lack the experience needed to feel the likelihood of probabilities.
To overcome this difficulty, it is necessary to use simulation in the teaching and
learning probability at school. This paper would like to introduce Monte Carlo
simulation which is a method of solving probability problems through the use of
experiments. This method should be used in teaching probability to high school
students because it increases their practical experience and it teaches students how
to apply probability when attempting to solve real world problems. This paper
explains what Monte Carlo simulation is and why and how teachers should teach
Monte Carlo simulation to high school students.

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JOURNAL OF SCIENCE OF HNUE
Interdisciplinary Science, 2014, Vol. 59, No. 5, pp. 64-70
This paper is available online at
USING MONTE CARLO SIMULATIONWHEN TEACHING PROBABILITY
TO HIGH SCHOOL STUDENTS
Nguyen Phuong Chi
Department of Mathematics Education, Hanoi National University of Education
Abstract. Probability is an important topic in the high school mathematics
curriculum. However, dealing with randomness is always a challenge for
Vietnamese students because they have not been encouraged to use their intuitive
abilities and they lack the experience needed to feel the likelihood of probabilities.
To overcome this difficulty, it is necessary to use simulation in the teaching and
learning probability at school. This paper would like to introduce Monte Carlo
simulation which is a method of solving probability problems through the use of
experiments. This method should be used in teaching probability to high school
students because it increases their practical experience and it teaches students how
to apply probability when attempting to solve real world problems. This paper
explains what Monte Carlo simulation is and why and how teachers should teach
Monte Carlo simulation to high school students.
Keywords: Monte Carlo simulation, model, experiment, teaching probability, high
school students.
1. Introduction
Probability is an appropriate topic in the school mathematics curriculum and
textbooks because it is an indispensable part of real life, it supplies necessary tools to
comprehend the world around us and it provides meaningful applications of mathematics
at all levels [1, 6, 9, 10, 11 and 13]. In addition, this topic can contribute to the
mental development of students [5, 6 and11] and it is inherently interesting, exciting,
and motivating for most students [13].
Although probability plays an important role in the teaching of mathematics at
school, it is always a difficult for students due to the cognitive demands of dealing
with randomness in contrast with the deterministic thinking associated with most uses of
Received November 05, 2013. Accepted June 19, 2014.
Contact Nguyen Phuong Chi, e-mail address: chinp@hnue.edu.vn.
64
Use Monte Carlo simulation in teaching probability for High school students
mathematics [8]. Coping with variability, samples, random trials, centers and distribution
is really a challenge for high school students because their ability to intuit is low and they
have not previously been shown that most of the real world is in the realm of calculable
probability.
One way to help students improve their ability to intuit and use probability is
to use simulation to provide them with an opportunity to obtain practical experience.
Simulation helps students understand how probability is applied and how to grasp real
life situations. Using simulation to teach probability at school is supported by educators
around the world. Shaughnessy, an American educator, recommended that the study
of probability in schools should rely on simulation to model experiments that require
the use of problem-solving techniques [12]. A Swiss educator named Inhelder said that
simulation helps students discover and develop probabilistic truths in realistic problems
[7]. Wolpers and Go¨tz, two German educators, stated that students must first experience
probability situations to be able to understand models and such experience can be best
achieved through the use of simulations [16]. Another American educator named Bryan
said that practical problems from the simplest to the most complex can be solved, or at
least approximate answers can be found, using simulation. He also said that simulation is
an ideal mechanism for providing the teacher with the opportunity to develop a systematic
progression from estimating probabilities to drawing conclusions and making inferences
[2].
A look at Vietnamese high school textbooks shows that there is no use of simulation
to create probabilistic situations ([3]). This absence of simulation in the textbook and
the teaching of probability deny students an opportunity to gain practical experience and
apply probabilistic knowledge in the real world. This is not in line with that which is
stated in the Education Law of Vietnam: ”Educational content has to relate to real life”
and “Teaching methods have to teach students the how to apply in real life the information
they have learned in school” [4].
The authors of this study hope that teachers will begin to use Monte Carlo
simulation in order to improve the teaching and learning of probability at the high school
level. The efficiency of this method has been shown by educators around the world ([2],
[7], [12], [14], [15], [16]).
2. Content
2.1. What is Monte Carlo simulation?
Consider the following problems in probability (these problems are based on
problems presented in Watkins [15] and Travers [14]):
Problem 1: Each box of milk contains one of seven kinds of toys. A student wants
to know how many boxes of milk he should expect to buy to get the entire set of toys.
65
Nguyen Phuong Chi
Since this is a difficult problem to solve analytically, the student decides to solve it using
simulation. The student puts the names of the seven toys on slips of paper and put the slips
into a box. He withdraws one slip of paper, writes down the name of the toy and replaces
the slip of paper. This is continued until the name of each toy is drawn. The number
of boxes ’purchased’ (slips drawn) is recorded. Repeating the process twenty times, it is
found that the average number of box that must be purchased is 18.7.
Problem 2:Minh thinks he doesn’t have to study prior to taking tests. He is willing
to take his chances. Suppose that he take a true-false test in Geography and doesn’t know
the answers to ten of the questions. What’s the chance that he’ll get seven or more of those
ten questions correct by guessing?
To solve this problem, he uses a coin with heads meaning “Minh has the correct
answer” and tails meaning “Minh has the wrong answer.” Minh tosses the coin ten times
and count, the number of times it came up heads. If it came up heads seven of more times,
he writes down “Minh has seven or more correct answers.” When this process is repeated
100 times, it is found that Minh has seven or more correct answers 21 of the 100 times.
Thus P (seven or more correct answers) is, for these 100 sets of tossed,
21
100
or 0.21 or
21%.
Problems 1 and 2 above have been solved by a technique called Monte Carlo
simulation. This is a method of solving probabilistic problems experimentally. It involves
finding a model for the given problem. The model is physically different, it is easier to
operate, and it has the same mathematical characteristics as the original problem.
The distinctive feature of Monte Carlo simulation is the use of objects such as dice
or coins. The theoretical basis for the Monte Carlo method is called the law of large
numbers, which states that the more times a simulation is run
(number of successes) / (number of runs)
The closer one gets to the actual analytical probability.
2.2. Why teach Monte Carlo simulation?
Monte Carlo simulation should be taught for the following reasons:
- First, as a type of simulation or mathematical model, this method teaches students
how to represent real-world systems in terms of mathematical relationship. One of the
most important aims of teaching probability is to help students learn how to solve various
problems in real life and Monte Carlo simulation can do it efficiently. Many real life
problems can be solved using Monte Carlo simulation. More specifically, almost all of
any probability or expected-value problems can be solved using an appropriate Monte
Carlo simulation [15]:
+ One basic type of problem involves determining the probability of success or
failure. For example, in problem 2 presented above, success to Minh is when he guesses
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Use Monte Carlo simulation in teaching probability for High school students
seven or more correct answers.
+ A second basic type of problem asks for an expected value, not a probability. For
example, in problem 1 presented above, students must answer the question: “How many
milk boxes he can expect to have to buy in order to get the entire set of toys?”
Therefore, knowing how to use Monte Carlo simulation can help students solve a
large class of real life problems. This method provides students with an opportunity to use
probability to understand certain aspects of real life situations.
- Second, Monte Carlo simulation is a useful way to verify the results obtained from
a purely analytic explanation. For example, if the probability that a baby will be a boy is
1
2
and the probability that the baby will be a girl is
1
2
, is the probability of having two boys
in a family of two children
1
3
or
1
4
? That is, is the sample space BB, BG or GG, or is it
BB, BG, GB GG? (B is boy, G is girl)
Students can be convinced that it is the latter by flipping two coins and suppose that
heads means ‘baby boy’, tails means ‘baby girl’. If this is done 100 times, a distribution
such as
HH HT TT
23 48 29
will appear, indicating that these three outcomes are not equally likely [15].
- Third, Monte Carlo is easy to do in the classroom because most of the materials
needed are at hand. Moreover, students find this kind of mathematics to be fun and
interesting and they enjoy trying to device new and clever variations [15].
2.3. How should Monte Carlo simulation be taught to high school
students?
2.3.1. Students should be shown and given the chance to practice the general steps
of Monte Carlo simulation
To use Monte Carlo simulation, teachers should teach students how to represent a
real world problem in terms of probability and then solve that problem. The best way is to
show students the general steps of Monte Carlo simulation so that they can follow these
steps when they face real life problem that are solvable using this method.
Travers [14] lists five general steps for the Monte Carlo approach to probability.
They are:
- Model: Find an appropriate model for the problem situation. For example, in
problem 2 the situation was modeled by using a coin, where heads means “Minh has a
correct answer” and tails means “Minh has a wrong answer”.
67
Nguyen Phuong Chi
- Trial: Determine what constitutes a trial consist of. Oftentimes a trial consists of
tossing the coin (or rolling the die) until a predetermined number of outcomes is obtained.
For example, in problem 2, a trial consists of tossing the coin ten times, once for each test
question.
- Successful trial: Determine whether or not a trial is successful. For example, in
problem 2, a successful trial occurs when the heads appears at least seven times, which
means seven or more correct answers are obtained.
- Number of trials: The trials are repeated until the predetermined number of trials
has been completed. For most problems undertaken in school, one hundred trials will
provide adequate accuracy. In problem 2, for example, one hundred trials should be
performed.
- Probability (success): Estimate the probability of a successful trial P (success) by
the ratio (number of successful trials) / (total number of trials). For example, in problem
2, the probability of getting seven or more correct answers is estimated by the ratio
21
100
.
These five steps need to be practiced as a problem solving technique by students
through various exercises. The demonstrative exercise is presented below:
Problem 3:What is the probability that in a group of four people chosen at random,
two or more were born in the same month?
Solution:
Model: Use a twelve-sided die (one side for each month of the year).
Trial: A trial consists of rolling the die four times, one for each person in the group.
Successful trial: A successful trial is one in which a number obtained more than
once in the four rolls of the die - that is, at least two people have the same birth month.
Number of trial: Repeat the trial at least 100 times.
Probability (success): The probability P that at least two people share the same
birth month is estimated by the ratio: number of success/number of trials.
2.3.2. Use Monte Carlo simulation with the support of computer software
Simulations can be rapidly produced with help of computer programs. Without
computer simulation, gathering sufficient experimental data to investigate problems
would be so time-consuming that it would not be feasible for the classroom [7].
At the high school level, Monte Carlo simulation can be performed more effectively
with the support of Excel software. Excel software is quite familiar to high school students
and this can help them simulate tossing a coin or rolling a die. With this software, the
students can toss a coin or roll a die thousands times representationally rather than actually
doing it with their hands.
For instance, to solve problem 4 above, instead of using a real twelve-sided die we
can model rolling this die with the support of EXCEL software as follow:
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Use Monte Carlo simulation in teaching probability for High school students
Simulation of rolling a twelve-sided die
In our case, a die has twelve faces. Hence, we use function RANDBETWEEN (1, 12).
This function will return a random integer between 1 and 12. To simulate 1000 rolls of
the die, we type A1= RANDBETWEEN (1, 12), for A2 to A1000 we only need to use
F4 or Ctrl-D to copy the formula of A1. Then we have column A which represents 1000
rolls of the twelve-sided die.
3. Conclusions
Using simulation in teaching and learning probability at school is a trend
worldwide. As a type of simulation, Monte Carlo simulation teaches students to model
and solve many real world problems. This method can help students by giving them
practical experience, improving their ability to experiment and helping them understand
how probability is applicable in real life. High school students should learn about and
practice the general steps of Monte Carlo simulation so that they can use this method
efficiently whenever they face a probabilistic problem. In addition, students should make
use of computer software such as Excel to perform the simulation more conveniently.
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