Gaming and probability are intertwined. In fact, there is an entire discipline in mathematics called “Game Theory.” In this assessment, you will apply discrete and continuous probability distributions to games. There will be two parts: the first will concentrate on discrete probability distributions, and the second part will focus on continuous probability distributions.

##### Instructions

###### Part One – Data Table

The theoretical probability of rolling a fair six-sided die is 1/6 for any specific single outcome, such as rolling a one. You want to test the theoretical probability by running an experiment.

In this experiment, you need to roll a six-sided die 25 times. Record the outcome of each die roll. Create a discrete probability distribution using your outcomes as the probability. For example, if you rolled 4 fives out of your 25 total rolls, your probability would be 4/25.

*x – 1 – 2 – 3 – 4 – 5 – 6**P(x)*

###### Part Two – Discrete Probability Distribution

After filling in the table above with your experimental probability, answer the following questions. Show all work for full credit. Calculations should be performed in Excel while answers including an explanation of steps using proper terminology are provided in a separate document.

1. What is the expected outcome for rolling a six-sided die using the discrete probability distribution table above?

2. What is the probability of rolling an even number according to the discrete probability distribution table above?

- How does this compare to the theoretical probability of 0.5?
- Explain why you think there is a difference between the theoretical probability and the experimental probability you found.

3. Create a binomial probability distribution based on the discrete probability distribution table above where a success is rolling an even number. Answer the following questions:

- How do you know this is a Binomial Probability Distribution? Explain by showing how this example fits all four properties of a Binomial Probability Distribution.
- Define
*n,p,q*. - What is the probability that you will roll exactly 12 even numbers?
- What is the probability that you will roll at least 12 even numbers?
- Find the expected number of even numbers that you will roll.

###### Part Three – Continuous Probability Distribution

Dice are a common tool used in several board games. One board game which utilizes two dice is Monopoly. While the outcomes of rolling two dice in this game would be a discrete random variable, we are interested in looking at a continuous random variable associated with Monopoly and its respective probability.

The time it takes to finish a game of Monopoly is normally distributed with a mean of 120 minutes and a standard deviation of 30 minutes. Using this premise, answer the following questions. Show all work for full credit. Calculations should be performed in Excel while answers including an explanation of steps using proper terminology are provided in a separate document.

- Explain why this is a continuous probability distribution instead of a discrete probability distribution.
- What is the probability that a game lasts less than 45 minutes?
- What is the probability that a game lasts more than 160 minutes?
- What is the z-score of a game that lasts exactly 105 minutes?