Engineering Probability Class 11 Thurs 2018-02-22

Source

1   The different counting formulae for selecting k items from n

  1. With replacement; order matters: \(n^k\).
  2. W/o replacement; order matters: \(n(n-1)\cdots(n-k+1) = \frac{n!}{(n-k)!}\).
  3. With replacement; order does not matter: \({{n-1+k} \choose k}\)
  4. W/o replacement; order does not matter: \({n\choose k}=\frac{n!}{k!(n-k)!}\).

2   Review questions

  1. Sampling with replacement with ordering: Each day I eat lunch at either the Union, Mcdonalds, Brueggers, or Sage. How many ways can I eat lunch over 5 days next week?
  2. sampling w/o replacement and w/o order: How many different possible teams of 3 people can you pick from a group of 5?
  3. sampling w/o replacement and with order: 5 people run a race for gold, silver, bronze. How many ways can the medals be won, w/o any ties?
  4. binomial: A coin falls heads with p=.6. You toss it 3 times. What's the probability of 2 heads and 1 tail?
  5. multinomial: You play 38-slot roulette 3 times. Once you got red, once black and once 0 or 00. What was the probability?
  6. conditional probability: You have 2 dice, one 6-sided and one 12-sided. You pick one of them at random and throw it w/o looking; the top is 2. What's the probability that you threw the 6-sided die?
  7. What's the expected number of times you'll have to toss the unknown die to get your first 2?
  8. Independence: Consider {1,2,3,...12}. Is the set of even numbers independent of the set of multiples of 3? What if we use {1,2,..10}?
  9. Useful review questions from the text.
    1. 2.83 on page 90.
    2. 2.99.
    3. 3.5 on page 130.
    4. 3.9.
    5. 3.15.
    6. 3.26.
    7. 3.88 on page 139.

3   Iclicker

  1. We often add a check bit to an 8-bit byte, and set it so there are an odd number of 1 bits. When we read the byte, which is now 9 bits, if there are an even number of 1 s, then we know that there was an error.

    Assume that the probability of any one bit going bad is 1e-10. (The real number is much smaller.)

    What is the probability of the byte going bad (within 1 significant digit)?

    1. 1e-10
    2. 8e-10
    3. 9e-10
    4. 3.6e-19
    5. 7.2e-19

  2. What is the probability of the byte going bad, but we don't notice that (because there were 2 bad bits)?

    1. 1e-10
    2. 8e-10
    3. 9e-10
    4. 3.6e-19
    5. 7.2e-19

4   Chapter 4 ctd

  1. Taxi example: Sometimes there are mixed discrete and continuous r.v.
    1. Let X be the time X to get a taxi at the airport.
    2. 80% of the time a taxi is already there, so p(X=0)=.8.
    3. Otherwise we wait a uniform time from 0 to 20 minutes, so p(a<x<b)=.01(b-a), for 0<a<b<20.
  2. Iclicker. For the taxi example, what is F(0)?
    1. 0
    2. .2
    3. .8
    4. .81
    5. 1
  3. iclicker. For the taxi example, what is F(1)?
    1. 0
    2. .8
    3. .81
    4. .9
    5. 1
  4. Simple continuous r.v. examples: uniform, exponential.
  5. The exponential distribution complements the Poisson distribution. The Poisson describes the number of arrivals per unit time. The exponential describes the distribution of the times between consecutive arrivals.
  6. The most common continuous distribution is the normal distribution.
  7. Conditional probabilities work the same with continuous distributions as with discrete distributions.
  8. Using Matlab: Matlab, Mathematica, and Maple all will help you do problems too big to do by hand. I'll demo Matlab since IMO more of the class knows it.
  9. Iclicker. Which of the following do you prefer to use?
    1. Matlab
    2. Maple
    3. Mathematica
    4. Paper. It was good enough for Bernoulli and Gauss; it's good enough for me.
    5. Something else (please email about it me after the class).