8/18/2019 Tutorial on probability

1/2

Tutorial-1

Subject - Probability, Statistics and Random Process(IC 210) 

Q1. The events A and B are mutually exclusive. Can they be independent?

Q2. If and ( ) 1/ 4, ( ) 1/ 3 then find ( | ) and ( | ). A B P A P B P A B P B A  

Q3. Show that

( | ) ( | ) ( | ) and ( ) ( | ) ( | ) ( ) P A B C P A B C P B C P A B C P A B C P B C P C   

Q4. Show that ( ) ( ) and ( )( ) ( ) ( )c c c c c c c c A B A B A A B A B A B B A  

Q5. Show that then ( ) ( ).c A B P A P B   

Q6. 1 2 3 1 2 3If , , .....is asequence of events such that ..... then show that A A A A A A

lim ( )k nk n

 P A P A

 

Q7. Show that ( ) ( ) ( )c P A P A B P A B  

Q8. Consider the experiment of tossing a fair coin repeatedly and counting the number of tossesrequired until the first head appears.

(a) Find the sample space of the experiment.

(b) Find the probability that the first head appears on the k th

 toss.

(c)Verify that P(S) = 1

Q9. Consider the binary communication channel shown in Figure. The channel input symbol X

may assume the state 0 or the state 1, and, similarly, the channel output symbol Y may assume

either the state 0 or the state 1. Because of the channel noise, an input 0 may convert to an output

1 and vice versa. The channel is characterized by the channel transition probabilities p0, q0, p1, q1 

defined by

0 1 0 0 0 0

1 0 1 1 1 1

( | ) ( | )

( | ) ( | )

 p P y x q P y x

 p P y x q P y x

 

Where x0, x1, y0 and y1 denote the events (X = 0), (X = 1), (Y = 0) and (Y = 1), respectively.

 Note that p0 + q0 = 1 = p1 + q1. Let P(x0) = 0.5, p0 = 0.1, and p1 = 0.2.

(a) Find P(y0) and P(y1).

(b) If a 0 was observed at the output, what is the probability that a 0 was the input state?

(c) If a 1 was observed at the output, what is the probability that a 1 was the input state?

8/18/2019 Tutorial on probability

2/2

(d) Calculate the probability of error Pe.

Q 10.A committee of 5 persons is to be selected randomly from a group of 5 men and 10 women.

(a) Find the probability that the committee consists of 2 men and 3 women.(b) Find the probability that the committee consists of all women.

Q11.Suppose that a laboratory test to detect a certain disease has the following statistics. Let

A = event that the tested person has the disease

B = event that the test result is positive

It is known that ( | ) 0.99 ( | ) 0.005c P B A and P B A and 0.1 percent of the population actually

has the disease. What is the probability that a person has the disease given that the test result is

 positive?

Q12. Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years, it

has rained only 5 days each year. Unfortunately, the weatherman has predicted rain fortomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the time.

When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the probability that it

will rain on the day of Marie's wedding?