Probability and Random Process

1
Probability and Random Process

• Case Study Solution
• Module 1

2
Sample Space

• Problem 1 Count the number of voice packets
  containing only silence produced from a group of
  N speakers in a 10-ms period.
• Solution Denote sample space by S then,
• S 0, 1, 2, , N
• Problem 2 A block is transmitted repeatedly over
  a noisy channel until an error-free block arrives
  at the receiver. Count the number of transmission
  required.
• Solution Denote sample space by S then,
• S 1, 2, 3, ,

3
Sample Space

• Problem 3 Measure the time between two message
  arrivals at a message center.
• Solution Denote sample space by S then,
• S t t ? 0 0, )
• where t denotes time.
• Problem 4 Measure the lifetime of a given
  computer memory chip in a specified environment.
• Solution Denote sample space by S then,
• S t t ? 0 0, )
• where t denotes time.

4
Events

• Problem 1 Write the values of events for
  problems in case study of sample space for
  following events
• No active packets are produced
• Fewer than 10 transmission are required
• Less than t0 seconds elapse between message
  arrivals
• The chip lasts for more than 1000 hours but fewer
  than 5000 hour
• Solution
• No active packets are produced, then
• A 0

5
Events

• 2. Fewer than 10 transmission are required
• A 1, 2, , 9
• 3. Less than t0 seconds elapse between message
  arrivals
• A t 0 ? t lt t0 0, t0 )
• 4. The chip lasts for more than 1000 hours but
  fewer than 5000 hour
• A t 1000 lt t lt 5000 (1000, 5000 )

6
Events

• Problem 2 Measure the lifetime of a given
  computer memory chip in a specified environment.
  Let the events A, B, and C be defined by A(5,
  ), B (7, ) and C (0,3. Describe these
  events in words. Find the events A B, A
  C, and A B and describe them in words.
• Solution
• A ( 5 , ) lifetime is greater than 5
• B ( 7 , ) Lifetime is greater then 7
• C ( 0 , 3 Lifetime is not greater than 3
• A B ( 7 , ) Lifetime is greater than 5
  and 7
• A C Lifetime is greater than 5 and not
  greater than 3
• A B ( 5 , ) Lifetime is greater than 5 or
  7

7
Counting Sample Points

• Problem 1 How many seven-digit telephone number
  are possible if the first number is not allowed
  to be 0 or 1?
• Solution
• The number of distinct 7- tuples 8 10 10
  10 10 10 10
• 8 ( 106 )
• where 8 in the first place indicates that for
  choosing first digit of telephone number we have
  only 8 options form 2 to 9.
• while for the remaining positions we have all 10
  digits option is available.

8
Counting Sample Points

• Problem 2 A deck of cards contain 10 red cards
  numbered 1 to 10 and 10 black cards numbered 1 to
  10. How many ways are there of arranging 20 cards
  in a row?
• Solution
• Number draws form a deck of 20 distinct card
  20!. If we suppose that the first card can be red
  or black, then there are 20 choices for the first
  draw the next draw must be from the 10 cards of
  other color the next form the 9 of the next
  color, and so on
• 20 10 9 9 8 8 2 2 1 1 2
  (10!)(10!)

9
Counting Sample Points

• Problem 3 How many distinct permutations are
  there of four red balls, two white balls and
  three black balls?
• Solution
• Total number of balls are 9 from which 4 are
  red, 2 are white and 3 are black, then according
  to theorem total permutation can be found like
  this

10
Counting Sample Points

• Problem 4 Show that
• Solution
• According to the theorem we can prove like this,

11
Probability of an event

• Problem 1 In case of problem 2 of case study
  counting sample points Suppose we draws the
  cards at random and lay them in a row. What is
  the probability that red and black cards
  alternate?
• Solution

12
Probability of an event

• Problem 2 A park has N raccoons of which 10
  were previously tagged. Suppose that 20 raccoons
  are captured. Find the probability that 5 of
  these are found to be tagged? Denote this
  probability by p(N).
• Solution
• Number of ways of picking 20 raccoons out of N
• Number of ways of picking 5 tagged raccoons out
  of 10 and 15 untagged raccoons out of N 10
• Continue

13
Probability of an event
14
Probability of an event

• Problem 3 You win a lottery if you correctly
  predict the number of six balls drawn from an urn
  containing balls numbered 1, 2,, 49, without
  replacement and without regard of ordering. What
  is the probability of winning if you buy one
  ticket?
• Solution
• Here the probability of winning

15
Additive Rules

• Problem 1 Show that the probability that
  exactly one of the events A or B occur is given
  by
• Solution
• PA P1 P2
• PB P2 P3
• where
• now form the Rule,
  we can get

16
Additive Rules

• Problem 2 Show that
• Solution Identities of this type are shown by
  the application of axioms. We begin by treating
  as a single event, then

17
Conditional Probability

• Problem 1 A nonsymmetrical binary communication
  channel is shown below. Assume inputs are
  equiprobable.
• Find the probability that the output is 0.
• Find the probability that input was 0 given that
  the output is 1. Find the probability that input
  is 1 given that the output is a 1.
• Solution Continues

18
Conditional Probability

• Solution Let X denote the input and Y the
  output.
• a)
• b)

19
Conditional Probability

• Problem 2 A computer manufacturer uses chips
  from three sources. Chips from source A, B and C
  are defectives with the probability .001, .005,
  .01, respectively. If randomly selected chip
  found to be defective , find the probability that
  the manufacturer was A that the manufacture was
  C.
• Solution
• Continue

20
Conditional Probability
21
Bayes Rule

• Problem 1 One of the two coins is selected at
  random and tossed. The first coin comes up heads
  with probability p1 and second coin with
  probability p2. What is the probability that coin
  2 was used given that heads occurred?
• Solution

22
Bayes Rule

• Problem 2 A ternary communication channel is
  shown below. Suppose that the input symbols 0, 1,
  and 2 occurs with probability ½, ¼, and ¼
  respectively. Suppose that 1 is observed as an
  output. What is the probability that the input
  was 0? 1? 2?
• Solution Continue

23
Bayes Rule

• Solution