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Engineering Probability Class 11 and Exam 1 Solution - Thu 2020-02-20

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First Exam

Name, RCSID: W. Randolph Franklin, frankwr

Rules:

  1. You have 80 minutes.
  2. You may bring one 2-sided 8.5"x11" paper with notes.
  3. You may bring a calculator.
  4. You may not share material with each other during the exam.
  5. No collaboration or communication (except with the staff) is allowed.
  6. Check that your copy of this test has all six pages.
  7. You may omit questions totalling 4 points. You must cross out the ones you omit, or we will assume that you omitted the last subquestion.
  8. When answering a question, don't just state your answer, prove it.

Questions:

  1. This is a question about smoking and lung cancer. C is the event that someone has cancer. S is the event that someone smokes. Assume that

    1. 10% of smokers get lung cancer. P(C|S)=.1
    2. 90% of lung cancers happen to smokers. P(S|C)=.9
    3. Assume that 20% of people smoke. P(S)=.2

    Questions:

    1. (2 points) What is P(C)?

      P(C&S) = P(C|S) P(S) = .02

      = P(S|C) P(C)

      so P(C) = .02/.9 = .0222

    2. (2 points) What is P(C|S')?

      P(C&S') = P(C) - P(C&S) = .0222 - .02 = .0022

      P(C|S') = P(C&S') / P(S') = .0022/.8 = .00275

      So smoking is correlated with cancer.

      Fun fact: In the 1950s, there were ads with doctors saying that smoking was good for you.

  2. You are trying to pass the very difficult course ECSE-3030. For each time you try, you pass with probability 1/2. Your chance of passing this time is independent of how many times you've tried.

    The random variable is the number of times you have to take the course until you pass for the first time. E.g., if you pass on the first time, this number is one.

    1. (2 pts) What's the relevant probability distribution?

      geometric.

    2. (2 pts) What's the expected number of times you will have to take the course?

    3. (2 pts) What's the standard deviation?

      \(\sqrt{2} = 1.414\)

  3. This question is about tossing a 20-sided fair die, faces labeled from 1 to 20.

    1. Event A is that a number up to 10 shows.
    2. Event B is that an odd number shows.
    3. Event C is that the number is in the set {2, 4, 6, 8, 10, 11, 13, 15, 17, 19}.

    Questions:

    1. (2 points) Are A and B independent? Don't just say, yes or no. Prove your answer.

      By enumerating the sets:

      P(A) = P(B) = P(C) = 1/2

      P(A&B) = P(A&C) = P(B&C) = 1/4 = 1/2 * 1/2

      P(A&B&C) = 0 != 1/2 * 1/2 * 1/2

      So each pair is an independent pair. But the triple is not independent.

    2. (2 points) Are A and C independent?

    3. (2 points) Are B and C independent?

    4. (4 points) Are A, B, and C independent?

  4. This question is about transmitting a signal over a noisy channel. The source transmits either 0 or 1. However, you receive one of three (not two) signals: A, B, or C.

    1. P(0)=.2, P(1)=.8
    2. P(A|0)=.8, P(B|0)=P(C|0)=.1
    3. P(A|1)=P(B|1)=.2, P(C|1)=.6

    Questions:

    1. (6 points) What are P(A&0), P(B&0), P(C&0)?

      P(A&0) = P(A|0)P(0). So, .16, .02, .02

    2. (6 points) What are P(A&1), P(B&1), P(C&1)?

      .16, .16, .48

      Note that the six add to 1.

    3. (6 points) What are P(A), P(B), P(C)?

      P(A) = .16+.16 = .32

      P(B) = .18, P(C) = .5

    4. (6 points) What are P(0|A), P(0|B), P(0|C)?

      P(0|A) = P(A&0)/P(A) = .16/.32 = 1/2

      P(0|B) = 1/9

      P(0|C) = .02/.5 = .04

  5. An LCD display has 2000 * 2000 pixels. A display is accepted if it has 10 or fewer faulty pixels. The probability that a pixel is faulty coming out of the production line is 1e-6.

    1. (2 points) What's the appropriate probability distribution for the number of bad pixels in a display?

      Poisson.

      The question, "What's the appropriate probability distribution", not what are possible distributions. I've made the point repeatedly that the hard part is knowing which math to use. Binomial is not appropriate here. Using it would require computing numbers like 4000000!.

    2. (2 pts) What's the mean number of bad pixels in a display?

    3. (2 pts) What's the probability that a display has all good pixels?

      \(e^(-4} 4^0 / 0! = e^(-4).\)

    4. (4 pts) What proportion of displays are accepted? An expression is ok; you don't need the actual number.

      \(\sum_{k=0}^{10} e^{-4} 4^k / k!\)

End of exam 1, total 50 points.