3. A soda machine has 10 buttons of which two never work, one works half the tim

Question

3. A soda machine has 10 buttons of which two never work, one works half the time and the rest work all the time. A coin is inserted and a button is pushed at random. What is the probability that no soda is received? If no soda is received, what is the probability that the button pushed is one of the ones that never works? a. b. c. If soda is received, what is the probability that the button pushed is the one that only works half the time 4. Consider the following binary symmetric channel with P(x]:06. P[X2-04, and p=0.1. 1-p X1 Yi Y2 1-p a. Find the probability that x, was transmitted given that y2 was received b. Find the probability that x2 was transmitted given that y2 was received c. Find the a priori probability of error. d. Find the a priori probability of receiving y

Explanation / Answer

Q1)

--- Part a)

P(No soda received) = (2/10)*(1) + (1/10)*(1/2) = 1/4

=> Answer = 1/4

--- Part b)

A = "button pushed in one of those which never works" and B = "no soda received"

Note: "|" denoted "GIVEN"

P(A | B) = P(B | A) * P(A) / P(B); P(B|A) = 1, P(A) = 2/10 and P(B) = 1/4

=> Answer = 4/5

--- Part c)

A = "button pushed in one which works half the time" and B = "Soda received"

P(A|B) = P(B|A)*P(A)/P(B) = (1/2)*(1/10)/(1-1/4) = 1/15

=> Answer = 1/15

----

Q2)

--- Part a)

A = "x1 was transmitted" and B = "y2 was received"

P(A|B) = P(B|A)*P(A)/P(B) = p*(0.6)/(0.6*p + (1-p)*(0.4)) = 0.06/(0.06 + 0.36) = 1/7

=> Answer = 1/7

--- Part b)

A = "x2 was transmitted" and B = "y2 was received"

P(A|B) = P(B|A)*P(A)/P(B) = (1-p)*(0.4)/(0.6*p + (1-p)*(0.4)) = 0.36/(0.06 + 0.36) = 6/7

=> Answer = 6/7

--- Part c)

A = "error in transmission"

P(A) = P(x1 was transmitted but y2 was received) + P(x2 was transmitted but y1 was received)

= (0.6)*(p) + (0.4)*p = p = 0.1

=> Answer = 0.1

--- Part c)

A = "y1 is received"

P(A) = P(x1 was transmitted and y1 was received) + P(x2 was transmitted and y1 was received)

= (0.6)*(1-p) + (0.4)*p = 0.54 + 0.04 = 0.58

=> Answer = 0.58

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