Chicago Game Probability

MTH109 – Mathematical Explorations

Colorado State University – Global

Dice games have been around for many, many years. They are games of chance and therefore require no skill. Many different ways of playing with dice have been invented, and one of those games is called Chicago.

Chicago is a simple game that requires 2 dice. There can be as many players as desired and there are 11 rounds that make up the game. The goal of the game is to earn the most points by the end of the game. To earn points you must roll both dice and the sum of the dice must equal the target combination of that round. Example: Round 1 has a target combination of 2. If you roll a combination that equals 2 then you earn 2 points. Round 2 has a target combination of 3; Round 3 is 4; Round 4 is 5; et cetera, et cetera.

This game does not require any skill, as stated before, and it is impossible to know if you will roll the right combination. Because this is a game of chance with a set amount of outcomes, it is possible to determine the probability of a roll. Knowing the probability of a roll during each round of Chicago can help determine the odds of winning the game.

**Knowing the Space**

In games of chance it helps to know the sample space and the event space. The sample space is the total number of outcomes (Monterey Institute), so in this particular game the total number of outcomes is 6*6 = 36. The way that we can create this space is by using a table with six columns and six rows. When laying out the sample space for rolling two dice at once, you are creating a diagram that shows all of the possible outcomes from each roll. If we have that diagram we can quickly determine the probability of certain rolls. To determine the percentage we would just need to follow our formula: Probability = Number of Desired Outcomes / Number of Possible Outcomes.

**Probability v****. Odds**

Probability and odds are often used interchangeably but are not at all the same thing. “The odds for an event is the ratio of the number of ways the event can occur to the number of ways it does not occur” (Wasserstein, 2014). Odds are always displayed in a “X to X” manner, for example the odds of a coin flip landing on heads is 1 to 1 or 1:1. Probabilities are stated in a different manner; as a percentage. “Probabilities should also be stated in the range zero to 100 percent” (Wasserstein, 2014). This would mean that the probability of a coin landing on heads in a coin flip is 50%.

**Independent Events**

Events are a big part of probability. In a dice game, in order to accurately determine the probability or odds it’s important to know if the event of rolling a die is independent or dependent. This means understanding if the probability of a roll changes based on other elements like more than one die. In question 2 of part one, rolling the target combination and rolling a one would both be independent events. This is because you are still rolling 2 dice to achieve a result. The reason rolling a one is an independent event is due to the fact that the result of one die is not dependant on the other.

**Complementary Events**

Complementary events are similar to independent events in probability. Complementary events are events that are similar. In question 3 of part 1 the events are complementary. These are complementary events because in both circumstances the player rolled a one. Regardless of what the other die shows, so long as the die showing a one remains constant in both rolls, the player will either win or lose.

**Game Changer**

The great thing about games of chance is that you can always up the challenge by adding rules. In question 4 of part 1 we are given examples of some possible rules to add to the fun. Those rules are if the player rolls the correct target combination then they will earn as many points as whatever the target combination happens to be. If they roll the right combination then they get a dollar for every point and if they don’t earn any points then the player loses 3 dollars.

Example: Bill and Sue are playing Chicago. They are on the third turn so the target combination is 4. Bill rolls a 2 and a 1, so he gets no points. This also means that he losses 3 dollars. Sue rolls next and rolls a 2 and a 2, so she gets 4 points and earns a dollar for every point. In the next turn, the target combination is 5. Bill rolls and his combination equals 5, so he wins 5 points and earns 5 dollars. Sue rolls and the combination equals 9, so she gets no point and must subtract 3 dollars from her winnings

**Advantages of Probability**

By knowing the probabilities when playing dice games you can increase the likelihood of winning. In this particular game, rolling for a total combination of 7 has the highest possible probability, but it is still fairly low. The likelihood that both players earn points on this turn is still higher than during any other turn.

There are lots of scenarios where probabilities can be used, it doesn’t just have to be dice games. Another scenario could be something a simple as a random grab out of a jar. Let’s say I have a tub of gumballs that has 25 pieces: 3 red, 5 blue, 4 orange, 6 purple, 4 green, 3 yellow. What’s the probability that I’ll grab a purple gumball without looking in the container? Because we know the total number of outcomes and how many of each color there is we can determine that the probability of me picking a purple gumball blindly out of a container is 24%.

**Conclusion**

Probability is something that can be used for many different applications. It is the likelihood of an event happening, and life is full of random phenomena, therefore it is something we can use to deal with uncertainty. A classic example is flipping a coin. There are two possible outcomes in this situation; the coin could come up heads or tails. Since we can’t know which will happen, we can say that there is a probability for each of the two events to occur. Probability isn’t the only way for expressing uncertainty, but it is probably the most important and the most widely used.

References

Monterey Institute. (n.d.). Probability. Retrieved October 19, 2019, from http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U08_L4_T1_text_final.html.

Wasserstein, R. (2014, December 14). Odds or Probability? Retrieved October 18, 2019, from https://senseaboutscienceusa.org/know-the-difference-between-odds-and-probability/.

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