ECE 313 Problem Set # 3 Fall 2002
Assigned: 9/11/02 Due: 9/18/02
Conditional probabilities and Bayes formula
Assigned reading: Ross Sections 3.1–3.3 and first five pages of Section 3.4.
Noncredit exercises: Chapter 3, problems 1,2,5,10,12,16,31,38,39,44
1. Randomly selected die
There are three dice in a bag. One has one red face, another has two red faces, and the third has three red
faces. One of the dice is drawn at random from the bag, each die having an equal chance of being drawn.
The selected die is repeatedly rolled. (a) What is the probability that red comes up on the first roll?
(b) Given that red comes up on the first roll, what is the conditional probability that red comes up on the
second roll?
(c) Given that red comes up on the first three rolls, what is the conditional probability that the selected die
has red on three faces?
2. Ternary channel
A communication system transmits three signals, s1, s2,or s3, with equal probabilities. The reception is
corrupted by noise, causing the transmission to be changed according to the following table of conditional
probabilities:
Receive, j
s1s2s3
s10.8 0.1 0.1
Send, i s20.05 0.9 0.05
s30.02 0.08 0.9
The entries list the probability of sjreceived, given that siis sent. For example, if s1is sent, the conditional
probability of receiving s3is 0.1. (a) Compute the probabilities that s1, s2, s3are received.
(b) Compute the probabilities P[sisent |sjreceived] for i, j = 1,2,3.
3. Matching problem
A router receives 8 packets that it knows need to be routed to eight stations A, B, C, D, E , F ,Gand H, each
station receiving exactly one packet. Due to a lightning strike at the router site, it cannot parse the header
information in a packet and hence it randomly assigns the packets to stations. So each station receives one
packet, but all possibilities for the assignment are equally likely.
(a) Find the probability that the router gets at least 6 packets routed correctly?
(b) Given that Station A does not receive the correct packet, what is the probability that none of the stations
receive their designated packets?
4. Frantic search
At the end of each day Professor Plum puts her glasses in her drawer with probability .90, leaves them on the
table with probability .06, leaves them in her briefcase with probability 0.03, and she actually leaves them
at the office with probability 0.01. The next morning she has no recollection of where she left the glasses.
She looks for them, but each time she looks in a place the glasses are actually located, she misses finding
them with probability 0.1, whether or not she already looked in the same place. (After all, she doesn’t have
her glasses on and she is in a hurry.)
(a) Given that Professor Plum didn’t find the glasses in her drawer after looking one time, what is the
conditional probability the glasses are on the table?
(b) Given that she didn’t find the glasses after looking for them in the drawer and on the table once each,
what is the conditional probability they are in the briefcase?
(c) Given that she failed to find the glasses after looking in the drawer twice, on the table twice, and in the
briefcase once, what is the conditional probability she left the glasses at the office?
