18.440 Probability and Random Variables
MIT Department of Mathematics
February 21, 2007
Due: Wednesday, February 28, 2007
Problem Set 3
Problem 1
A new test has been developed to determine whether a given student is overstressed.
This test is 95% accurate if the student is not overstressed, but only 85% accurate if the
student is in fact overstressed. It is known that 99.5% of all students are overstressed.
Given that a particular student tests negative for stress, what is the probability that the
test results are correct, and that this student is not overstressed?
Problem 2
Chapter 3, Problem #23
Problem 3
Chapter 3, Problem #58
Problem 4
Chapter 3, Problem #76
Problem 5
Chapter 3, Theoretical Exercise #7
Problem 6
Chapter 3, Theoretical Exercise #24
Problem 7
Consider the n × n integer grid S = {(i, j) : 1 ≤ i ≤ n, 1 ≤ j ≤ n}. Color each point of S
red with probability 21 independently at random, and color the remaining points blue. A
square in this grid is a set of four points {(a, b), (a + `, b), (a, b + `), (a + `, b + `) : a, b, ` ∈
N } ⊆ S. A square is monochromatic if all four points have the same color. What is the
expected number of monochromatic squares?
1