The game of craps is played with two dice. The player rolls both dice and wins immediately if the outcome (the sum of the faces) is 7 or 11. If the outcome is 2, 3, or 12, the player loses immediately. If he rolls any other outcome, he continues to throw the dice until he either wins by repeating the first outcome or loses by rolling a 7.
a) Explain how to simulate the roll of a single fair die. (Hint: Just use digits 1 to 6 and ignore the others.) Then explain how to simulate a roll of two fair dice.
b) Draw a tree diagram for one play of craps. In principle, a player could continue forever, but stop your diagram after four rolls of the dice. Use Table A, beginning at line 114, to simulate plays and estimate the probability that the player wins.
Your company operates a van service from the airport to downtown hotels. Each van carries 7 passengers. Many passengers who reserve seats don’t show up – in fact., the probability is 0.2 that a randomly chosen passenger will fail to appear. Passengers are independent. If you allow 9 reservations for each van, what is the probability that more than 7 passengers will appear? Do a simulation to estimate this probability.
A famous example in probability theory shows that the probability that at least two people in a room have the same birthday is already greater than 1⁄2 when 23 people are in the room. The probability model is:
- The birth date of a randomly chosen person is equally likely to be any of the 365 dates of the year.
- The birth dates of different people in the room are independent.
To simulate birthdays, let each three-digit group in Table A stand for one person’s birthday date. That is 001 is January 1 and 365 is December 31. Ignore leap years and skip groups that don’t represent birth dates (Hint only numbers from 001-365 are to be considered). Use line 139 of Table A to simulate birthdays of randomly chosen people until you hit the same date a second time. How many people did you look at to find two with the same birthday?With a computer, you could easily repeat this simulation many times. You could find the probability that at least 2 out of 23 people have the same birthday, or you could find the expected number of people you must question to find two with the same birthday. These problems are a bit tricky to do by math, so they show the power of simulation.