COSC 6221: Statistical Signal Processing Theory
Assignment # 1: Review of Probability
Due Date: September 23, 2003
1. A coin with probability of head given by p{H }  p  (1  q) is tossed n times. Calculate the probability
that the number of heads that appear are even.
2. A shipment contains K good and ( N  K ) defective components. We pick at random (n  K )
components and test them. Show that the probability p that k of the tested components are good
equals
 K  N  K 

p   
 k  n  k 
N
 
n
3. If P( A)  0.6 and k is the number of successes of A in n trials,
a. show that P{550  k  650}  0.999 , for n  1000 .
b. Find n such that P{0.59 n  k  0.61n}  0.95.
4. A coin is tossed an infinite number of times. Calculate the probability that k heads are observed at
the n ’th toss but not earlier.
5. For an average driver, the probability of suffering an accident in 1 month is 0.02. Show that the
probability that in 100 months an average driver will have exactly 3 accidents is given by
P  1.333 e 2
6. Players X and Y roll a pair of dice alternatively starting with X . The player that rolls eleven wins.
Show that the probability p that X wins equals 18/35.
7. In the ternary (3-ary) communication channel shown in fig.1,
Fig. 1: A 3-ary communication channel
the probability of 3 being sent is three times more frequently than a 1 and a 2 being sent is twice
more frequently as 1. If a 1 is observed, what is the conditional probability that a 1 is transmitted?
8. In a ring network consisting of 8 links as shown in fig. 2, there are two paths connecting any terminals
denoted by `o’. Assume that the links fail independently with probability, q, 0  q  1. Find the
probability of a successful transmission from terminal A to terminal B. Note that terminal A transmits
the packet in both directions on the ring. Also, terminal B removes the packet from the ring upon
reception. Successful transmission means that the terminal B received the packet from at least one of
the two possible directions.
Fig. 2: Ring topology used in computer networks.
9. A random-number generator generates integers from 1 to 9 (inclusive). All outcomes are equally
likely; each integer is generated independently of any previous integer. Let  denote the sum of two
consecutive generated integers, i.e.,   N1  N 2  .
a. Given that  is odd, what is the conditional probability that  is 7.
b. Given that  > 10, what is the conditional probability that  will be odd.
10. Assume that a faulty receiver produces audible clicks to the great annoyance of the listener. The
average number of clicks per second depends on the receiver temperature and is given by
  1  exp   / 10 
where  is the time from turn on. Derive a formula for the probability of 0, 1, 2, … clicks during the first
10 seconds of operation during turn-on. Assume the Poisson law.