Download Assignment 2

MTHE/STAT 351 — Fall 2016
Homework Assignment 2
due Monday, Oct. 3, in class
Chapter, section and problem numbers refer to the 3rd edition of the Ghahramani textbook.
Four of the following six problems will be chosen at random to be marked.
1. A number x is selected at random from the interval [−2, 3].
(a) Consider the events A = {x < − 12 }, B = {|x| < 1}, and C = {x ≥
probabilities P (A), P (B), P (C), P (AB), P (AC), and P ((A − B) − C).
1
}.
2
Find the
(b) Find the probability that x will be a solution of the equation x2 − x = 2.
2. Section 2.2, # 18.
3. Section 2.2, # 23.
4. A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is yellow.
(a) If the child puts the blocks in a line, how many different arrangements are possible?
(b) If one of the arrangements in part (a) is randomly selected, what is the probability that
no two black blocks are next to each other.
5. Review Problem # 16, page 73.
6. Consider a 4 × 6 grid as shown in the figure below.
(6,4)
(0,0)
You are to take a path along the grid beginning at the point (0,0) and ending at the point
(6,4), but you are only allowed to move upwards or to the right. One such path is highlighted
in the figure.
(a) How many such paths are possible?
[Hint: How many “ups” and how many “rights” do you need in such a path.]
(b) If you were to randomly pick such a path, what is the probability that you will pass
through the center of the grid, i.e., the point (3,2)?
Bonus question: We are given n (n > 5) points in three dimensional space (R3 ) that are in
general position, i.e., no four of them lie on the same plane. Let Ω denote the set of the planes
determined by any three of these points and assume that no two planes in Ω are parallel so that
any two planes intersect in a straight line. If such a straight line is chosen at random, what is the
probability that it does not pass through any of the n points?
(Bonus questions do not have to be attempted, but bonus marks will be awarded for a correct solution.)