# Random Variables

Problem Statement:

The game of Unlucky 7 is one of the most famous (or notorious) of all gambling games played with dice. In this game, the player rolls a pair of six-sided dice (assumed to be fair), and the SUM of the numbers that turn up on the two faces is noted. If the sum is 7 or 11 on the first roll, then the player wins immediately. If the sum is 2, 3, or 12 on the first roll, then the player loses immediately. If any other sum appears, then the player continues to throw the dice until the first sum is repeated (the player wins) or the sum of 7 appears (the player loses).

Requirements:

1. Compute the theoretical probabilities for each sum of a dice roll (rounded to the nearest thousandth decimal place). Use for the simulation model.
• Since the random variable is the sum of the two fair die, we need to create a probability table that correlates to the probability of the sums 2-12, inclusive.
• From prior knowledge, I know that this particular probability distribution is binomial, and the total number of ways is 6² because there is six possibilities on a fair dice, and since we are rolling two at the same time, it is the exponent.
• The decimal approximation for P(SUM = 7) is rounded down one thousandths place so the sum of all the decimal approximation values are equal to one, and not above one.
• Hence, now the total of the decimal approximations are 1.
1. Design a five-step simulation experiment for the game of Unlucky 7. Describe each of the five steps completely. Conduct 20 trials. Record the results.
• Problem:
1. To simulation 20 trials of the game Unlucky 7.
• Key Components:
1. Upon the first roll, if the player receives a 7 or an 11, then they win immediately. If they receive a 2, 3, or 12, then they lose immediately.
2. If any other sum appears than the ones listed above, the player must keep on playing until the first sum is repeated, in which they ...