E155B - Stanford University



Mathematical and Computational Methods for Engineers

E155C, Winter 2004

Problem Set #1

(Sample Spaces, Experiments, Probability)

Date: 1/7/2004 Due: 1/14/2004

A man who travels a lot was concerned about the possibility of a bomb on board his plane. He determined the probability of this, found it to be low but not low enough. Now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board would be miniscule.

Reading: Sample spaces, events: Ross 3.1-3.3

Axioms of probability: Ross 3.4

Counting, permutations, combinations: Ross 3.5

MATLAB Workbook: Exercise 1

Problem 1 Based upon relative frequency, it has been determined that 15% of all cars tested emit excessive hydrocarbons, 12% emit excessive CO, and 8% emit excessive amounts of both. Let E and F denote the events that a randomly selected car emits excessive hydrocarbons and CO, respectively. Express the following events in terms of E and F and find the appropriate probabilities:

a) emissions of both hydrocarbons and CO are excessive

b) at least one emission is excessive

c) neither emission is excessive

d) hydrocarbon emission is not excessive

e) hydrocarbon emission is excessive, but CO emission is not

Problem 2 Three light bulbs are chosen at random from 15 bulbs of which 5 are defective. Find the probability that:

a) none are defective

b) exactly one is defective

c) at least one is defective

Problem 3 Ten identical computers are subjected to excessively hot operating conditions. Based on past experience, there is 20% chance of failure for any given computer. Find the probabilities of the following events:

a) none fail

b) all fail

c) at least one fails

d) exactly one fails

Problem 4 Two points a and b are selected at random along the x-axis such that [pic] and [pic]. Find the probability that the distance between a and b is greater than 3.

Problem 5 A point is selected at random inside a circle. Find the probability that the point is closer to the center of the circle than to its circumference.

Problem 6 Craps is a popular gambling game played in casinos throughout the world. The player rolls two dice and plays against the house. If the first roll is 7 or 11, the player wins immediately; if it is 2, 3, or 12, the player loses immediately. If the first roll results in 4, 5, 6, 8, 9, or 10, the player continues to roll until either the same number appears, which constitutes a win, or a 7 appears, which results in the player loosing. What is the player’s probability of winning the game ?

Problem 7 Observations of random phenomena are best described and classified by means of a histogram – a plot of frequencies of occurrence of all possible outcomes. Suppose that two fair dice are thrown and the sum of the numbers of dots is recorded. It is desired to determine, by means of a “virtual” experiment, the likelihoods of occurrence of the sums of these two integers. Write a MATLAB script to simulate 10,000 trials of the experiment and make a histogram plot of the resulting frequencies. [Hint: you may find MATLAB functions unidrnd and hist useful]

Problem 8 The goal of this problem is to determine whether or not it is more likely to get at least one double six in 24 throws of a pair of dice or to get at least one six in 4 throws of a single die. Since the analytical methods for solving a problem of this type have not yet been discussed, your task is to solve this problem numerically by conducting a set of “virtual” experiments using MATLAB. [Hint: use the unidrnd function to generate a vector of random integers ranging from 1 to 6. Then, use a for loop to perform the same experiment a sufficient number of times to determine the asymptotic behavior of the relevant relative frequencies]

Problem 9 Using MATLAB, write a script to numerically simulate the game of craps in Problem 6. Follow the suggestions outlined in the Hint of Problem 8 and perform 1,000,000 runs. Compare your numerical answer to the analytical result in Problem 6.

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