Introduction
Yubisuma is a traditional number guessing game played in Asian countries such as Singapore. I used to play this too when I lived in Singapore as a child. The main objective of the game is to guess the number of thumbs, which the participants will raise. There is no limit to the number of players that can join the game. To start, the fists must be closed and placed together, inner wrist to inner wrist. Considering two players, one player will take a guess on the number of thumbs that will be raised by both players. For this case, a number from 0 to 4 can be chosen. Upon selecting the number, the player will say carefully “Yubisuma”, and the number of his or her choice. At the same moment that the number was stated, both players must be able to raise or not raise one or raise both or none of their thumbs. The player wins if he guessed the correct number of thumbs raised. Using probability and statistics, the tendency to win in the game is investigated.
 Statement of Task
 Determine the effect of the following game mechanics to the probability or chances of winning in “Yubisuma” game.
 Effect of the number of thumbs/fingers that can be raised by two players.
 One thumb for each player.
 Two thumbs for each player.
 Four fingers for each player.
 Effect of the number of players (2, 3, 4, and 5 players) each using two thumbs only.
 Rationale
Yubisuma game is expected to get harder as the number of participants increases, since the number of available guesses or options increases as well. Same is correct if the number of fingers to use increases.
 Plan of Investigation
 Probability Distribution
For the case in which each player can use one hand and thus one thumb only, the value that can be obtained is either 0, 1, or 2 only. A tree diagram is constructed for the two players to determine the frequency of appearance of the stated values. Then the probability of obtaining a certain number or value is computed.
Probability is defined as a measure of the chance of a random occurrence. It describes the longterm proportion with which a certain event will occur in situations with shortterm uncertainty. Sample space lists all possible outcomes or events (Dekking, et al., 2005).
The probability of an event E, denoted P(E), is the likelihood of that event taking place, wherein the value must be (Dekking, et al., 2005):
Using the classical method, probability of an event is calculated as (Dekking, et al., 2005):
Where is the number of ways an event E can occur, and is the total number of possible outcomes.
Histogram is prepared to show the distribution of the possible guessed value by the player. The number value will be the discrete independent variable and will be plotted against its probability to create the probability distribution curve (Wilkinson, 1998).
 Computation of Mean and Standard Deviation
In order to find the mean () of the given distribution, the following formula is used (Milton & Arnold, 2002):
Each value of is multiplied by each probability , then add all results.
The standard deviation () of the given distribution is computed using the formula (Milton & Arnold, 2002):
Using the mean value computed earlier.
 Mathematical Investigation
A tree diagram is prepared based on the stated changes in the mechanics of the game in order to determine the probability of an event to take place. For this case, the event pertains to the number of fingers raised by all the participants of the game, while the sample space is all of the possible numbers that can be guessed by the player. So that is equivalent to the combined number of fingers allowed to use for all participants plus 0 for the case that none was raised. The probability of an event to take place is computed as:
 Effect of the number of thumbs/fingers that can be raised by two players.
Table 1. Probability computation for varying number of fingers to use by two players only
Condition  Tree Diagram  Frequency Table  
A. 2 players each can use a single thumb. Maximum number: 2

 
B. 2 players each can use both thumbs. Maximum number: 4 
 
F. 2 players each can use four fingers. Maximum number: 8 

The mean and standard deviation of the discrete variables or the number and its probability value is computed using the given formula for the mean:
And standard deviation:
Table 2. Comparison of computed mean and standard deviation for varying number of fingers to use by two players only
Condition  Max Value  Mean Value  Std Dev  Prob Mean 
A  2.00  1.00  0.50  0.71 
B  4.00  2.00  0.33  1.15 
F  8.00  4.00  0.20  2.00 
Using the frequency distribution data, a histogram is prepared with a fitted curve to show the distribution of the events and its probability. Using condition B, the probability distribution is shown below. Notice that it follows a uniform or bellshaped distribution which is the ideal case. It clearly shows that using the mean value, the probability is higher compared to the values less than or greater than the mean.
Note that by using the mean value as the guessed number, the player has a 33.33% chance to win the game. This is caused by the number of times the mean value can occur during the game.
Figure 1. Probability distribution using the normal game mechanics with two players each using both thumbs only (B)
Figure 2. Effect of number of fingers to use (A – 2, B – 4, F – 8) by 2 players to the probability of winning
Based on figure 2, the probability of winning using the mean value decreases as shown by the peak of each histogram. The graph becomes wider as the number of fingers to use increases. This indicates more options or events to take place thus the probability became more distributed. Thus, the game becomes tougher as the number of available fingers to use by two players increases.
 Effect of the number of players (2, 3, 4, and 5 players) each using two thumbs only
Using the same procedure in 5.1. wherein a tree diagram was constructed, the frequency of each event is determined and summarized in Table 3. Figure 3 shows the probability distribution using the histogram.
Table 3. Frequency table for varying number of players each using both thumbs only
B (2P)  P(B)  C (3P)  P(C)  D (4P)  P(D)  E (5P)  P(E) 
0.0000  0.1111  0.0000  0.0370  0.0000  0.0123  0.0000  0.0041 
1.0000  0.2222  1.0000  0.1111  1.0000  0.0494  1.0000  0.0206 
2.0000  0.3333  2.0000  0.2222  2.0000  0.1235  2.0000  0.0617 
3.0000  0.2222  3.0000  0.2593  3.0000  0.1975  3.0000  0.1235 
4.0000  0.1111  4.0000  0.2222  4.0000  0.2346  4.0000  0.1852 
5.0000  0.1111  5.0000  0.1975  5.0000  0.2099  
6.0000  0.0370  6.0000  0.1235  6.0000  0.1852  
7.0000  0.0494  7.0000  0.1235  
8.0000  0.0123  8.0000  0.0617  
9.0000  0.0206  
10.0000  0.0041 
For each case of different numbers of players, the mean value and the standard deviation was computed.
Table 4. Effect of Varying Number of Players
Players  Mean  Probability  SD 
2  2  0.33  1.15 
3  3  0.26  1.41 
4  4  0.23  1.63 
5  5  0.21  1.83 
Figure 3. Probability Distribution Plot of Number Guessed for Varying Number of Players (B – 2, C – 3, D – 4, E – 5)
Figure 4. Relation of Probability of the Mean Value Vs the Number of Players
As shown in figure 3, the plot becomes wider as the number of players increases. Increasing the number of players each being able to use two thumbs only, increases the number of possible events to occur. Such events correspond to the number of alternative values that can be used by the player. Thus, increasing the number of players would result in fewer chances of winning using the mean value as an initial guess. Thus the play gets tougher as the number of players increases.
Conclusion
The mathematical investigation proves that the game becomes tough as the number of players increases. Increasing the number of players increases the number of available options of guess values for the players. This results in a wider distribution of probabilities as shown in the histogram plot. Same is true when the number of allowed fingers to use increases. Even by having two players only, as the number of allowed fingers increases, the number of options increases as well, leading to the same outcome as when the number of players increases.
References
Dekking, F. M., Lopuhaa, H. P., Kraaikamp, C. & Meester, L. E., 2005. A Modern Introduction toProbability and Statistics. London: SpringerVerlag.
Milton, S. J. & Arnold, J. C., 2002. Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences. 4th ed. s.l.:McGrawHill Higher Education.
Wilkinson, D. J., 1998. MAS131: Introduction to Probability and Statistics. [Online]
Available at: https://www.staff.ncl.ac.uk/d.j.wilkinson/teaching/mas131/notes.pdf
[Accessed 25 September 2016].