ECE331 HOMEWORK 2


(0)[reading corresponding with the lectures] 
Read Yates chapter 1.5,1.6,1.7,1.9,2.1,2.2,2.3,2.4.

(1) Yates 1.5.2
You have a six sided die that you roll once.
Let R_i denote the event that the roll is i.
Let G_j denote the event that the roll is greater than j.
Let E denote the event that the roll of the die is even-numbered.
(a) What is P[R_3|G_1], the conditional probability that 3
is rolled given that the roll is greater than 1?
(b) What is the conditional probability that 6
is rolled given that the roll is greater than 3?
(c) What is P[G_3|E], the conditional probability that the roll is
greater than 3 given that the roll is even?
(d) Given that the roll is greater than 3, what is the 
conditional probability that the roll is even?  

(2) Yates 1.5.5
You have a shuffled deck of three clubs: 2, 3, and 4
and you deal out the three cards.
Let E_i denote the event that the ith card dealt is even-numbered.
(a) What is P[E_2|E_1], the probability that the 
second card is even given that the first card is even?
(b) What is the conditional probability that the first two cards are even
given that the third card is even?
(c) Let O_i denote the event that the ith card dealt is
odd-numbered.  What is P[E_2|O_1], the conditional probability that the
second card is even given that the first card is odd?
(d) What is the conditional probability the
second card is odd given that the first card is odd?

(3) Yates 1.6.5
In an experiment with equiprobable outcomes,
the sample space is S={1,2,3,4} and P[{s}]=1/4 for all s in S.
Find three events in S that are pairwise independent 
but are not independent.

(4) Yates 1.6.7 
For independent events A and B, prove that 
(a) A and B^c are independent
(b) A^c and B are independent
(c) A^c and B^c are independent
[NOTATION: A^c means A complement]

(5) Yates 1.7.4
You have two biased coins. 
Coin A comes up heads with probability 1/4.
Coin B comes up heads with probability 3/4.
However, you are not sure which is which so you 
choose a coin randomly and you flip it.
If the flip is heads, you guess that the flipped coin is B;
otherwise you guess that the flipped coin is A.
Let events A and B designate which coin was picked.
What is the probability P[C] that your guess is correct?

(6) Yates 1.7.5
Suppose that for the general population, 1 in 5000 people 
carries the human immunodeficiency virus (HIV).
A test for the presence of HIV yields either a positive (+)
or a negative (-) response.
Suppose that the test gives the correct answer 99% of the time.
What is P[-|H], the conditional probability that a person tests 
negative given that the person does have the HIV virus?
What is P[H|+], the conditional probability that a randomly chosen person
has the HIV virus given that the person tests positive?