# Probability density

1,)  Craps is an interesting casino game because it is an example of a random experiemtn that takes place in stages; the evolution of the game depends critically on the outcome of the first roll. In particular, the number of rolls is a random variable.

The player (known as the shooter) rolls a pair of fair dice

a. If the sum is 7 or 11 on the first throw, the shooter wins; this event is called a natural.

b. If the sum is 2, 3, or 12 on the first throw, the shooter loses; this event is called craps.

c. If the sum is 4, 5, 6, 8, 9, or 10 on the first throw, this number becomes the shooter’s point. The shooter continues rolling the dice until either she rolls the point again (in which case she wins) or rolls a 7 (in which case she loses).

The sum of the scores  on a given roll has the probability density function in the table. The table of bets (not relevant to this problem) is shown in the image.

Setup an experiment to simulate the sum of a single roll of two 6-sided dice.

Run 1000 replications of the experiment.

Analyze the percentage of time the shooter (1) wins on the first roll or (2) loses on the first roll. (Note: we are ignoring item C above for this problem).

Summarize the sample of the SUMS in a frequency distribution, probability distribution, cumulative probability distribution.

Graph the results.

What would the theorectical probability of winning or losing on the first roll based on classical assignment of probabilities? Graph?

2.) Roulette is the oldest casino game still in operation. It’s invention has been variously attributed to Blaise Pascal, the Italian mathematician Don Pasquale, and several others. In any event, the roulette wheel was first introduced into Paris in 1765. Here are the characteristics of the wheel:

The (American) roulette wheel has 38 slots numbered 00, 0, and 1–36.

Green slot numbers 0, 00;

Red slot numbers 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36;

Black slot numbers 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35

The wheel is spun and then a small ball is rolled in a groove, in the opposite direction as the motion of the wheel. Eventually the ball falls into one of the slots.

All 38 possible values for a single spin of the wheel are given in the table. Naturally, we assume mathematically that the wheel is fair, so that the random variable X that gives the slot number of the ball is uniformly distributed over the sample space. An image of the roulette wheel and it’s betting table (not relevant to this problem) is shown in the image.

Setup an experiment to simulate the outcome for a single spin of the wheel.

Run 1000 replications of the experiment.

Analyze the percentage of time the a player wins

1) who only bets on the same number for every spin of the wheel

2) who randomly selects a different number every spin

Describe your findings in the management statement.

** PLEASE SEE ATTACHED FOR BOTH QUESTIONS**

Files: HW02.xlsx