The document analyzes the probability distribution of landing on different sides when tossing a traditional Mongolian game piece called a shagai. Data was collected from tossing a shagai 200 times and recording the results. Statistical analysis found the probability of landing on sheep or goat sides was highest at around 0.385, while probability of camel and horse were lower at 0.095 and 0.135 respectively. The distribution was modeled using a three-parameter function estimated through the method of moments and maximum likelihood estimation. Confidence intervals were also calculated based on results from 50 trials of tossing shagai 20 times each.

1. WHAT IS THE DISTRIBUTION FUNCTION
OF “SHAGAI” TRADITIONAL GAME?
STATISTICS AND NUMERICS
GMIT – BACHELOR II students
TEAM 2: Bat-Ochir , Indra , Enkhnomin, Battsengel, Naranbileg
11/25/18

2. ABSTRACT
This paper analyzes what is the distribution function of “shagai” traditional game. To answer
this question, we made trials which was tossing one shagai and noticing how the shagai was
landed.
We measured 200 samples and calculated descriptive statistics such as mean, variance and
illustrated proper figure. In order to find out parameters of function, we used Maximum
likelihood estimator, Moment of method and calculated confidence interval for the average
probability.
Our results showed the function with three parameters and it shows that the probability of
landing sheep and goat are greater than horse and camel.

3. INTRODUCTION
Shagai is one of the Mongolian traditional games. Shagai generally land on one of its four
sides: horse, camel, sheep and goat. A fifth side-cow is possible on uneven surface and was not
observed in our trial.
The purpose of this assignment was to determine the distribution function of shagai, also called
probability distribution function. The probability of landing on one side is not same in all four
sides. Because the outer surface of shagai is not even and has different areas.
In our prediction, the probability distribution of tossing shagai is biased because gravity and area
of landing surface directly affects the probability.

4. SAMPLE PROTOCOL
The raw data is collected by the trial of tossing shagai. Shagai is tossed 200 times and
each result is listed on excel. Tossing shagai has four outcomes (horse, camel, sheep and
goat). In data they are converted into numbers for further calculation. For example: sheep
x=0, goat x=1, camel x=2, horse x=3.
For confidence interval calculation, the different trial was needed to be performed. Shagai is
tossed 20 times and the probability of each outcome is calculated. This trial is conducted 50
times. At the end, we got 50 values of probability for each outcomes.
Trial 1:
161 G 1
162 G 1
163 S 0
164 H 3
165 G 1
166 G 1
167 S 0
168 H 3
169 G 1
170 C 2
171 S 0
172 G 1
173 S 0
174 G 1
175 S 0
176 C 2
177 G 1
178 G 1
179 S 0
180 S 0
181 S 0
182 H 3
183 S 0
184 S 0
185 S 0
186 S 0
187 G 1
188 S 0
189 S 0
190 G 1
191 S 0
192 S 0
193 S 0
194 G 1
195 G 1
196 G 1
197 C 2
198 G 1
199 G 1
200 G 1
121 G 1
122 G 1
123 H 3
124 S 0
125 S 0
126 S 0
127 S 0
128 S 0
129 S 0
130 S 0
131 G 1
132 G 1
133 C 2
134 S 0
135 G 1
136 G 1
137 S 0
138 G 1
139 G 1
140 G 1
141 G 1
142 S 0
143 S 0
144 G 1
145 S 0
146 S 0
147 G 1
148 H 3
149 S 0
150 G 1
151 G 1
152 S 0
153 S 0
154 C 2
155 S 0
156 S 0
157 S 0
158 S 0
159 G 1
160 G 1
1 H 3
2 G 1
3 S 0
4 C 2
5 H 3
6 G 1
7 G 1
8 G 1
9 G 1
10 G 1
11 C 2
12 S 0
13 G 1
14 G 1
15 S 0
16 C 2
17 H 3
18 S 0
19 S 0
20 S 0
21 G 1
22 S 0
23 G 1
24 S 0
25 G 1
26 S 0
27 S 0
28 G 1
29 C 2
30 G 1
31 G 1
32 G 1
33 C 2
34 H 3
35 H 3
36 S 0
37 H 3
38 G 1
39 G 1
40 G 1
41 C 2
42 G 1
43 S 0
44 G 1
45 H 3
46 G 1
47 H 3
48 H 3
49 C 2
50 S 0
51 S 0
52 S 0
53 S 0
54 H 3
55 C 2
56 C 2
57 S 0
58 G 1
59 G 1
60 H 3
61 S 0
62 S 0
63 G 1
64 S 0
65 S 0
66 H 3
67 G 1
68 C 2
69 H 3
70 G 1
71 S 0
72 G 1
73 G 1
74 S 0
75 G 1
76 G 1
77 C 2
78 H 3
79 S 0
80 S 0
81 G 1
82 G 1
83 G 1
84 S 0
85 S 0
86 S 0
87 H 3
88 S 0
89 G 1
90 H 3
91 G 1
92 H 3
93 S 0
94 G 1
95 S 0
96 H 3
97 S 0
98 G 1
99 H 3
100 G 1
101 C 2
102 H 3
103 H 3
104 G 1
105 G 1
106 C 2
107 G 1
108 S 0
109 S 0
110 C 2
111 S 0
112 S 0
113 S 0
114 G 1
115 S 0
116 H 3
117 G 1
118 S 0
119 S 0
120 G 1

5. Trial 2:
Trial
Number Sheep Goat Camel Horse
1 0.3 0.4 0.15 0.15
2 0.25 0.5 0.1 0.15
3 0.3 0.25 0.2 0.25
4 0.4 0.35 0.1 0.15
5 0.35 0.4 0 0.25
6 0.4 0.3 0.15 0.15
7 0.45 0.45 0.05 0.05
8 0.55 0.35 0.05 0.05
9 0.35 0.45 0.1 0.1
10 0.5 0.4 0.05 0.05
11 0.45 0.4 0 0.15
12 0.35 0.5 0.05 0.1
13 0.45 0.35 0.15 0.05
14 0.25 0.3 0.25 0.2
15 0.55 0.35 0.05 0.05
16 0.25 0.5 0.15 0.1
17 0.3 0.45 0.2 0.05
18 0.35 0.5 0.1 0.05
19 0.55 0.4 0.05 0
20 0.6 0.2 0.2 0
21 0.45 0.45 0.05 0.05
22 0.55 0.3 0.1 0.05
23 0.4 0.35 0.25 0
24 0.4 0.3 0.15 0.15
25 0.3 0.55 0.05 0.1
26 0.35 0.55 0.1 0
27 0.3 0.55 0.1 0.05
28 0.4 0.55 0.05 0
29 0.4 0.45 0.1 0.05
30 0.4 0.35 0.1 0.15
31 0.4 0.5 0.1 0
32 0.4 0.35 0.1 0.15
33 0.4 0.4 0.15 0.05
34 0.5 0.35 0.1 0.05
35 0.45 0.5 0 0.05
36 0.5 0.45 0 0.05
37 0.45 0.2 0.25 0.1
38 0.25 0.45 0.15 0.15
39 0.45 0.3 0.1 0.15
40 0.4 0.5 0 0.1
41 0.4 0.35 0.1 0.15
42 0.4 0.35 0.2 0.05
43 0.55 0.25 0.1 0.1
44 0.4 0.5 0.1 0
45 0.65 0.15 0.15 0.05
46 0.4 0.5 0 0.1
47 0.25 0.7 0 0.05
48 0.45 0.3 0.2 0.05
49 0.4 0.5 0.05 0.05
50 0.5 0.25 0.1 0.15
Mean 0.41 0.401 0.103 0.086
S.D 0.095298 0.109958 0.068817 0.063116

6. ANALYSIS AND RESULT
As a result of the trial of tossing shagai 200 times, the distribution of each outcome is
determined. According to the data, whereas the probabilities of landing on sheep and goat are
both 0.385, the probabilities of landing on camel and horse are relatively low, 0.095 and
0.135, respectively.
Frequency Relative Frequency Percent
Sheep 77 0.385 39%
Goat 77 0.385 39%
Camel 19 0.095 10%
Horse 27 0.135 14%
For 5 times larger sample (tossing 1000 times), the distribution was slightly different. The
possibility of rolling a sheep is the highest 0.41 but the possibility of rolling a horse is only
0.09. It can be seen that distribution of rolling shagai is biased. Shagai is more likely to land
on sheep or goat comparing to the camel and horse. It is often believed that horse is lucky side
of shagai. Surprisingly, our data suggests that camel has lowest probability instead of horse.

7. PARAMETER ESTIMATION
We assumed the variable X is the sides of shagai. For example: X is equal to 0, when
shagai lands on its sheep side and so on (goat x=1, camel x=2, horse x=3). The probabilities
of four outcomes are discrete and completely independent. So we consider, the probability of
rolling sheep, goat, and camel is equal to a, b and c, respectively.
Since the sum of all probability must be zero, we can say that probability of rolling horse is 1-
a-b-c. Our distribution function has three unknown parameters and we can find them using
two different parameter estimation methods.
X 0 1 2 3
(Sheep) (Goat) (Camel) (Horse)
P(X) a b c 1-a-b-c

8. a. METHOD OF MOMENTS
3 − 3𝑎 − 2𝑏 − 𝑐 =
1
𝑛
∑ 𝑥𝑖 = 0.98
9 − 9𝑎 − 8𝑏 − 5𝑐 =
1
𝑛
∑ 𝑥𝑖
2
= 1.98
27 − 27𝑎 − 26𝑏− 19𝑐 =
1
𝑛
∑𝑥𝑖
3
= 4.79
{
𝑎 = 0.385
𝑏 = 0.385
𝑐 = 0.095
1 − 𝑎 − 𝑏 − 𝑐 = 0.135
Population Sample
𝐸[ 𝑥] = ∑ 𝑥 ∙ 𝑃(𝑥) 𝜇1 =
1
𝑛
∑ 𝑥 𝑖
𝐸[ 𝑥2] = ∑ 𝑥2
∙ 𝑃(𝑥) 𝜇2 =
1
𝑛
∑ 𝑥 𝑖
2
𝐸[ 𝑥3] = ∑ 𝑥3
∙ 𝑃(𝑥) 𝜇3 =
1
𝑛
∑ 𝑥 𝑖
3
1
𝑛
∑ 𝑥 𝑖 0.98
1
𝑛
∑ 𝑥 𝑖
2
1.98
1
𝑛
∑ 𝑥 𝑖
3
4.79
X 0 1 2 3
P(X) a b c 1-a-
b-c

9. b. MAXIMUM LIKELIHOOD
𝐿( 𝑎, 𝑏, 𝑐) = ∏ 𝑓( 𝑥) = 𝑎77
𝑏77
𝑐19(1− 𝑎 − 𝑏 − 𝑐)27
𝑛
𝑘=𝑖
𝑙𝑛𝐿 = 77 𝑙𝑛𝑎 + 77 𝑙𝑛𝑏 + 19 𝑙𝑛𝑐 + 27(1 − 𝑎 − 𝑏 − 𝑐)
𝑑𝑙𝑛𝐿
𝑑𝑎
=
77
𝑎
−
27
1 − 𝑎 − 𝑏 − 𝑐
= 0
104𝑎 = 77 − 77𝑏 − 77𝑐
𝑑𝑙𝑛𝐿
𝑑𝑏
=
77
𝑏
−
27
1 − 𝑎 − 𝑏 − 𝑐
= 0
104𝑏 = 77 − 77𝑎 − 77𝑐
𝑑𝑙𝑛𝐿
𝑑𝑐
=
19
𝑐
−
27
1 − 𝑎 − 𝑏 − 𝑐
= 0
46𝑐 = 19 − 19𝑎 − 19𝑏
{
𝑎 = 0.385
𝑏 = 0.385
𝑐 = 0.095
1 − 𝑎 − 𝑏 − 𝑐 = 0.135

10. CONFIDENCE INTERVAL
In order to determine confidence interval, we used data from second trial. 50 probability
values are recorded on excel file. The mean and standard deviation value is calculated and
now we can find confidence interval.
Since n>30, following formula is used
Confidence interval for the average probability of rolling a sheep:
Trial number=n=50
Z-score =𝑧(𝛼/2) = 1.96 (95%)
[1.65 for 90%, 2.58 for 99%]
Sample mean = 𝑥̅ =0.4100
Sample standard deviation = s = 0.095298
(0.41) ± (1.96) × (0.095298) / (√50) = 0.4100 ± 0.0264
0.3836 ≤ 𝜇 ≤ 0.4364

11. CONCLUSION
The result of our experiment shows that distribution of shagai is biased and each
probability is completely independent. The probability of rolling sheep or goat always higher
than camel or horse.
Distribution function of shagai has three parameters a, b and c. Parameter estimation methods
like Method of moments and Maximum likelihood can be applied. As a result, three
parameters are calculated (a=b=0.385 c=0.095).
The confidence interval of any outcome can be calculated. As an example, the confidence
interval for the probability of rolling sheep was determined and it was suggested that average
probability of sheep in interval of 0.3836 ≤ 𝜇 ≤ 0.4364.

12. APPENDIX
TRIAL 1 - SHAGAI (200) - Method of Moments.xlsx
TRIAL 2 - SHAGAI (1000) - Confidence Interval.xlsx
Shagai.pptx