Statistical Analysis of Outcomes in Card Decks: A Mathematical Exploration

Abstract

In this study, we explore the mathematical concepts of tree diagrams, probability, expectation, combination, and standard deviation to analyze the outcomes of card draws from decks of 52 and 104 cards. This analysis provides insights into the probabilities of drawing number or picture cards in successive draws, with an emphasis on understanding how these probabilities and expected values change when the deck size is doubled. The investigation also delves into the calculation of total outcomes using combinations and examines the spread of results using standard deviation.

Introduction

The mathematical exploration of card decks through statistical and probabilistic methods offers a unique perspective on predicting outcomes and understanding the dynamics of card games, such as blackjack. By analyzing decks of 52 and 104 cards, we aim to uncover the differences in outcome probabilities and to deduce the implications of these differences on game strategies and decision-making processes.

Tree Diagram and Probability Analysis

The tree diagram is used for displaying all the possible outcomes of the event.

In this case, this tree diagram will represent possible outcome whenever the dealer draws the next card. Below is the example of the first result then the second result of 52 cards and 104 cards with replacement since both tries does not have a picture card (P/P).

P(Event of situation)=(N(Number of outcomes))/(N(Total number of outcomes))

P(E)=(N(E))/(N(s))

P(N,N)(N,N)=3/5×2/5=6/25

P(N,N)(N,P)=3/5×3/5=9/25

P(N,P)(N,N)=2/5×2/5=4/25

P(N,P)(N,P)=2/5×3/5=6/25

P(N,N)(N,N)=2/5×2/5=4/25

P(N,N)(N,P)=2/5×3/5=6/25

P(N,P)(N,N)=3/5×2/5=6/25

P(N,P)(N,P)=3/5×3/5=9/25

In the deck of 104 cards has more chances to get mostly a number with the picture rather than or getting both numbers in both tests meanwhile a deck with 52 cards shows more chances of getting a pair of numbers on the first test then a number with a picture at the second test.

Expectation Calculation

By using expectation, we can get predicted result from the data that already gathered. The most possible number outcomes in the data only have 3 chances of getting both numbers (N/N), both pictures (P/P) or a number pair with a picture card (N,P). The result of expectation is done in decimal.

Total of the outcomes = 3

Expected value=Σ(each possible outcome×the probability outcome)

E=Σ(N×P)

E=[1×P(N,N)]+[(2×P(N,P)]+[(3×P(P,P)]

P(N,N)= total result of number with number from the graph

P(N,P)= total result of number with picture from the graph

P(P,P)=total result of picture with picture from the graph

52 cards

P(N,N)=13/25

P(N,P)=10/25

P(P,P)=2/25

E=[1×(13/25)]+[(2×(10/25)]+[(3×(2/25)]

E=[0.52]+[(0.8]+[0.24]

E=1.56

104cards

P(N,N)=10/25

P(N,P)=12/25

P(P,P)=3/25

E=[1×(10/25)]+[(2×(12/25)]+[(3×(3/25)]

E=[0.4]+[(0.96]+[0.35]

E=1.71

After done with the calculation, a deck with 104 cards has higher than better probability with expectation value rather than a deck with 52 cards by 0.15 since the data collected with the random variable the result will vary.

Combination for Total Outcomes

Combination is used for calculating the total outcomes for the events since the pattern is not important. Below will be showing a formula will be used and calculation for the total outcome for one deck of cards and two deck of cards:

C_r^n=n!/r!(n-r)!

n!=the factorial of the total number of items

r!=the factorial of the number of items chosen

! is called factorial, the function of this symbol is to multiply the whole number backwards from the chosen number down to 1.

One deck of cards

n(The total number of items)=52

r(The number of items chosen)=2

C_2^52=52!/2!(52-2)!

C_2^52=1326

Two decks of cards

n(The total number of items)=104

r(The number of items chosen)=2

C_2^104=104!/2!(104-2)!

C_2^104=5356

The result shows two decks of cards have high value for the total outcomes than one deck of cards, since the total number of items or cards in two decks of cards has more compare to one deck of cards.

Standard Deviation for Outcome Distribution

The use of standard deviation for my exploration is to determine if the numbers are spread out or close to average. Before directly calculate the standard deviation for both decks of cards, I need to calculate the value of mean first then put all the value inside the table below in order to simplify the calculation.

x ̅=(Σ(x∙F))/ΣF

x ̅ (mean)=(Σ(result value∙Frequency))/ΣFrequency

One deck of cards

Σ(x∙F)=(1∙3)+(2∙2)+(3∙4)+(4∙1)+(5∙3)

Σ(x∙F)=38

ΣF=3+2+4+1+3

ΣF=13

x ̅=38/13

x F x∙F (x-x ̅ ) (x-x ̅ )^2

1 3 3 -1.92 3.70

2 2 4 -0.92 0.85

3 4 12 0.08 0.01

4 1 4 1.08 1.16

5 3 15 2.08 4.33

Σ(x-x ̅ )^2=3.70+0.85+0.01+1.16+4.33=10.05

σ (Standard Deviation)=((The total of 〖(Result value-Mean)〗^2)/(Σ(frequency)))^(1/2)

σ=((Σ(x-x ̅ )^2)/ΣF)^(1/2)

σ=(10.05/13)^(1/2)

σ=0.87

Two decks of cards

Σ(x∙F)=(1∙2)+(2∙2)+(3∙3)+(4∙1)+(5∙2)

Σ(x∙F)=29

ΣF=2+2+3+1+2

ΣF=10

x ̅=29/10

x F x∙F (x-x ̅ ) (x-x ̅ )^2

1 2 2 -1.90 3.61

2 2 4 -0.90 0.81

3 3 9 0.10 0.01

4 1 4 1.10 1.21

5 2 10 2.10 4.41

Σ(x-x ̅ )^2=3.61+0.81+0.01+1.21+4.41=10.05

σ=((Σ(x-x ̅ )^2)/ΣF)^(1/2)

σ=(10.05/10)^(1/2)

σ=1.00

Above of the calculation does approve that two decks of cards are more spread out than one deck of cards with a range of 0.13 standard deviation value. What I have notice here that both of data at Total of (x-x ̅ )^2 are the same with the total of 10.05, this mean that the Total of (x-x ̅ )^2 can be similar number throughout different number of decks.

Conclusion

What I have gathered by doing this investigation, In the deck of 104 cards has more chances to get mostly a number with picture (N,P) meanwhile a deck with 52 cards shows more chances of getting a pair of numbers (N,N) for the total result. A deck with 104 cards has a higher value in the probability of getting a pair of pictures (P,P) compare to a deck with 52 cards with different of 1/25, also a deck of 104 cards has 1.71 of expectation which means it is higher than a deck with 52 cards which has 1.56 of expectation. The result from the calculation of standard deviation of these two-deck showing two decks of cards has higher value with 1.00 than one deck of cards 0.87, this mean that two decks of cards are more separate than a deck of cards.

In conclusion, random variable does occur and change in probability every time the decks are shuffled. Since there are random variables are involved in blackjack, the data and the value shown will vary by repeating the data collection. Both expectation values are not high since the deck always be shuffled. The result of the calculation of standard deviation of the data show a close result with a range of 0.13, this value is to compare how different of the equal separation for the deck for distribute any possible cards.