.. title: Engineering Probability Class 11 Thurs 2018-02-22
.. slug: class11
.. date: 2018-02-21
.. tags: mathjax
.. category: class
.. link: 
.. description: 
.. type: text

.. sectnum::
.. contents:: Table of contents
..

    
The different counting formulae for selecting **k** items from **n**
--------------------------------------------------------------------

#. With replacement; order matters:  :math:`n^k`.

#. W/o replacement; order matters:  :math:`n(n-1)\cdots(n-k+1) = \frac{n!}{(n-k)!}`.

#. With replacement; order does not matter:  :math:`{{n-1+k} \choose k}`

#. W/o replacement; order does not matter:  :math:`{n\choose k}=\frac{n!}{k!(n-k)!}`.


Review questions
----------------

#. 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?

#. sampling w/o replacement and w/o order: How many different possible teams of 3 people can you pick from a group of 5?

#. 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?

#. binomial: A coin falls heads with p=.6.  You toss it 3 times.  What's the probability of 2 heads and 1 tail?

#. multinomial: You play 38-slot roulette 3 times.  Once you got red, once black and once 0 or 00.  What was the probability?

#. 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?

#. What's the expected number of times you'll have to toss the unknown die to get your first 2?

#. 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}?

#. Useful review questions from the text.   

   a. 2.83 on page 90.
   #. 2.99.
   #. 3.5 on page 130.
   #. 3.9.
   #. 3.15.
   #. 3.26.
   #. 3.88 on page 139.

   


Iclicker
--------

#. 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)?
  
   a. 1e-10
   #. 8e-10
   #. 9e-10
   #. 3.6e-19
   #. 7.2e-19

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

   a. 1e-10
   #. 8e-10
   #. 9e-10
   #. 3.6e-19
   #. 7.2e-19

      
Chapter 4 ctd
-------------

#. Taxi example: Sometimes there are mixed discrete and continuous r.v.   

   a. Let X be the time X to get a taxi at the airport.
   #. 80% of the time a taxi is already there, so p(X=0)=.8.
   #. Otherwise we wait a uniform time from 0 to 20 minutes, so p(a<x<b)=.01(b-a), for 0<a<b<20.


#. Iclicker.   For the taxi example, what is F(0)?

   a. 0
   #. .2
   #. .8
   #. .81
   #. 1

#. iclicker.   For the taxi example, what is F(1)?

   a. 0
   #. .8
   #. .81
   #. .9
   #. 1

#. Simple continuous r.v. examples: uniform, exponential.
#. 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.
#. The most common continuous distribution is the **normal** distribution.
#. Conditional probabilities work the same with continuous distributions as
   with discrete distributions.

#. *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.
#. Iclicker.  Which of the following do you prefer to use?

   a. Matlab
   #. Maple
   #. Mathematica
   #. Paper.  It was good enough for Bernoulli and Gauss; it's good enough for me.
   #. Something else (please email about it me after the class).




Material added after class
--------------------------

#. `My handwritten tablet notes <../../handwritten/222.pdf>`_.