• Communications example: “Data communication along the drill string using acoustic waves”


    Paper by S. Sinanovic, D. Johnson, V. Shah and W. Gardner: http://ieeexplore.ieee.org/document/1326975/
    A new telemetry method in oil well services uses compressional acoustic waves to transmit data along the drill string to the surface.
    Normal drilling operations produce in-band acoustic noise at intensities comparable to the transducer output while lossy propagation
    through the drill string and surface noise further degrade the signal.
    A single receiver system has a capacity of several hundreds bits per second.
    Model and signal analysis developed in Matlab, verified in field measurements

  • Lecture 5 Conditional Probability


    Warm up question:

    Suppose you are on a game show, and you are given the choice of three doors:
    Behind one door is a car; behind the others, goats.
    You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat.
    He then says to you, "Do you want to pick door No. 2?"

    Is it to your advantage to switch your choice?


    Monty Hall problem


    The problem is known as Monty Hall problem
    You should always switch! (Counterintuitive for most people)
    If you switch, probability of winning the car is 2/3
    If you don’t switch, probability is 1/3

    Conditional probability


    P(B | A) is the probability of event B occurring, given that event A has already occurred.
    In other words, probability of event B happening is conditional on event A already occurring.
    Vertical line | in P(B | A) denotes the conditional probability

  • Conditional probability: motivating examples from digital communications


    Example: Digital communications channel has an error rate of 1 bit per received 10000 bits.
        Errors are rare, but sometimes tend to occur in bursts, such as scratches on a DVD or poor wireless channel for mobile communications.
    That is why it is observed that if a bit is in error, the probability that the next bit is also in error in such cases is greater than 1/10000.

    Conditional probability: formal definition


    The conditional probability of an event B given an event A, denoted as P(B | A), is:
         P(B | A) = P(A ∩ B) / P(A)
         for P(A) > 0
    Since A is known to have occurred, A becomes the new sample space replacing the original sample space S.

    Conditional probability: relative frequency perspective


    From a relative frequency perspective of equally likely outcomes:
        P(A) = (number of outcomes in A) / (number of outcomes in S)
        P(A ∩ B) = (number of outcomes in A ∩ B) / (number of outcomes in S)
        P(B | A) = (number of outcomes in A ∩ B) / (number of outcomes in A)

    Conditional Probability: Example


    Find the probability that a single toss of a die will result in a number less than 4 if it is given that the toss resulted in an odd number.

    Click here for the solution  

    Another solution: Two out of three odd numbers are smaller than 4 (i.e. 1 and 3 are smaller than 4).
    Now, it is easy to see that this approach gives probability of having a number smaller than 4, given that the toss resulted in an odd number, equal to 2/3

    Conditional Probability: Example - Solution


    Solution: Let A be event that toss results in an odd number. A={1,3,5}, so P(A)=1/2
        • Let B be event that toss results in a number smaller than 4. B={1,2,3}, so P(B)=P(1)+P(2)+P(3)=1/6+1/6+1/6=1/2
        • A ∩ B={1,3} so P(A ∩ B)=1/3.
        • P(B|A)=P(A ∩ B) / P(A) =(1/3)/(1/2)=2/3

  • Conditional probability: example


    The probability that the first stage of an industrial process meets specifications is 0.9.
        The probability that it meets specifications in the second stage, given that it met specifications in the first stage, is 0.8.
        Determine the probability that both stages meet specifications.

    Click here for the solution  

    Conditional probability: previous example continued


    Since probabilities are usually below 1, multiplying increasing number of probabilities gives ever smaller number.
    If there were more stages of industrial process, the probability that the whole process met specifications would be very small
    This observation has important practical implications for quality control, cost and the industrial process in general

    Example: two bits


    Communication device transmits bits. Each bit is equally likely to be 0 or 1.
        If two bits are transmitted and one of them is equal to 1, find the probability that the other bit is equal to 1.

    Click here for the Solution: two bits (relative frequency approach)  

    Click here for the Solution: two bits (from conditional probability definition)  

    Conditional probability: example - Solution


    Let A and B denote the events that the process has met first and second stage specifications.
        The probability that both are met: P(A ∩ B) = P(B | A)·P(A) = 0.8·0.9 = 0.72

    Solution: two bits (relative frequency approach)


    There are four two bit transmissions:
        00,01,10, and 11. However, given that one of the bits is 1, only three two-bit outcomes are possible: 01, 10 and 11.

        Out of these three, only one will have both bits equal to 1. The answer is then 1/3.

    Solution: two bits (from conditional probability definition)


    Communication device transmits bits.
        Each bit is equally likely to be 0 or 1.
        If two bits are transmitted and one of them is equal to 1, find the probability that the other bit is equal to 1.

    Let A be event that both bits are equal to 1
        (A={11}) and let B be event that one of the bits
        is 1 (B={01,10,11}. Then, P(A∩B)=P(A).

        P(A)=1/4 and P(B)=3/4

        P(A|B)=P(A∩B)/P(B)=(1/4)/(3/4)=1/3
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