Blackjack: The Probability of Winning with Various Numbers of Decks

Probability of the outcomes in blackjack with various number of decks Introduction Blackjack is a simple trading card game that low chance to get a first winning hand. This trading card game is known by a lot of people mostly are teenagers and gamblers, but now it is not too common to hear in public cause, not every people are into it and also it is in the category of gambling and available in the casino. This game usually consists a minimum of 5 to 7 players, there is no age restriction for playing time game, teenagers play this game is to experience the excitement or thrilling feeling when playing blackjack. The length of this game does not eat a lot of time in less than 5 minutes the game already at the second game or third game.

During this investigation calculating the probability in the different amount of deck by using a basic rule of the game. Blackjack has a simple rule to follow: Dealer stands on soft 17 Late surrender is allowed Splitting pairs are acceptable Four hands can be split by players A separated card is placed after 4 and a half deck In this investigation, I will use two different amount cards with different decks and then find the probability of the outcomes in the game and I will be comparing both decks result. The aim that I am exploring this is to makes me understand more in probability problems and how useful in other situations.

My reflection about this makes me more active to do the research and makes me a quick learner. Formula Tree diagram First result × Next result P(E)=(N(E))/(N(s)) P(E)=P(Event of situation) N(E)=N(Number of outcomes) N(s)=N(Total number of outcomes) Expectation E(x)=∑_(i=1)^N▒ E(x)=Σ(N×P) E(x)=Expected value Σ(N×P)=(each possible outcome×the probability outcome) Data collection I will be collecting two data, each data will be repeated 5 times and using a different amount of deck. The deck will be shuffled before taking the data, it is to make sure that the deck is mixed equally. To avoid confusion, I’m using a red colour deck of cards for only one deck with the total of 52 cards, meanwhile I’m using a blue colour deck of cards as two decks with the total of 104 cards.

In the table I will be representing Jack, Queen and King as P or picture then the rest will be called number as N. Ace is including as a picture and has the value of 10 similar to Jack, Queen, and King. P= Picture N= Number Test of 52 cards (one deck) 1 P/N N/N P/N N/N N/N 2 N/N P/N N/P P/N N/N 3 N/N P/N N/N N/N N/N 4 P/N P/N N/P P/N N/N 5 P/P N/N P/P N/N N/N Test of 104 cards (two decks) 1 N/N N/N P/N N/P N/P 2 P/N P/N N/N N/N P/P 3 N/N N/N P/N P/N N/N 4 N/N P/N P/P N/P P/N 5 P/P N/N P/P N/P N/N The result of 52 cards (one deck) test N/N N/P P/P 1 3/5 2/5 0/5 2 2/5 3/5 0/5 3 4/5 1/5 0/5 4 1/5 4/5 0/5 5 3/5 0/5 2/5 TOTAL OF THE RESULT N/N N/P P/P 13/25 10/25 2/25 The result of 104 cards (two decks) test N/N N/P P/P 1 2/5 3/5 0/5 2 2/5 3/5 0/5 3 3/5 2/5 0/5 4 1/5 3/5 1/5 5 2/5 1/5 2/5 TOTAL OF THE RESULT N/N N/P P/P 10/25 12/25 3/25 By seeing the table above, I have noticed that the total of the probability to get two pictures of cards (P/P) in a deck of 104 cards is higher than a deck with 52 cards with different of 1/25, and also the probability to get both number cards (N/N) in 52 cards has higher number of probabilities which is 13/25 meanwhile in a deck of 104 cards has 10/25.

MATHEMATICAL TREE DIAGRAM

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

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.

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. 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.

Bibliography

https://www.bicyclecards.com/how-to-play/blackjack/ https://www.pagat.com/banking/blackjack.html https://www.beatblackjack.org/en/rules/
https://ibmathsresources.com/2013/10/12/the-gamblers-fallacy-and-casino-maths/ http://pi.math.cornell.edu/~mec/2006-2007/Probability/Blackjack.htm https://math.tutorvista.com/statistics/tree-diagram.html http://www.savvysoft.com/basic.htm
https://revisionmaths.com/advanced-level-maths-revision/statistics/expectation-and-variance