 Tues. Jan. 16
 Outline of the course. Some elementary probability.
If you roll two dice, what is the likelihood of at least one face
being a 2? What is the likelihood of the sum being 5?
Birthday Problem:
There are 25 people in a room. What is the likelihood that at least
two of them have the same birthday? What is the probability P that
exactly two of them have the same birthday? The new idea in this
problem is that computing probabilities involving
or is usually much more complicated than those
using and . We compute the probability Q that none
of the 25 people have the same birthday. Then P = 1  Q.
One reference for this lecture is Chapters 1 and 3 of
M. Grinstead, Charles M. and J. Laurie Snell, Introduction to
Probability. See
Math 210
Bibliography for pdf files with these chapters. If you find these
useful, you may want to buy the printed book.
 Wed. Jan. 17
 Writing your own web "Home Page". How do you give others
permission to read a web page you wrote? Where do you put these pages
on your computer?
Viewing pages others have created by using View Page Source on your
Web Browser.
A Simple Web Page
A More Complicated Example.
A Few Unix commands
(to be added: copying files and moving them to other directories)
 Thurs. Jan. 18
 Carrying out the computation in the Birthday Problem. We need to
compute (364/365)(363/365)...(341/365). This can be painful. We write
a computer program using perl
Birthday
Computation program for 50 people
If you roll one die 4 times, what is the probability of getting at
least one 6?
If you roll two dice 17 times, what is the probability of getting at
least one pair of 6's?
 Tues. Jan. 23
 Permutations: Say you have the letters A, B, C, D, and E.
How many different three letter "words" can you make using each letter
exactly once? Thus here the order of the letters is essential; the
word "BAD" is not the same as "DAB".
General Answer: If you have n "letters" there are n!/(nk)!
words with exactly k letters. For this example there are 5!/(53)! =60.
words.
Indistinct Objects: The same question, only now the order of the
k letters is irrelevant. Since there are 3! ways of rearranging three
letters, now there are only 60/3! = 10 sets with three different letters.
General Answer: If you have n "letters" there are
n!/[k!(nk)!] sets with exactly k letters. This is is the binomial
coefficient:
If you expand (1+x)^{n} = 1 + nx + ... , this is the coefficient of
x^{k}.
Binomial Probabilities: A biased coin is tossed five
times. Say there is a probability p of heads and thus q=(1p) of tails.
Say you win if exactly three of the tosses give heads. Thus you win
with any the sequences HHHTT, HHTHT, etc.
There is a probability of p^{3}q^{2} of the
winning sequences HHHTT, and similarly for any of the other winning
sequences HHTHT, etc. Since from the previous paragraph
there there are 5!/[3!(53)! = 10 different winning sequences, the
probability of winning is 10p^{3}q^{2}.
In general with n trials, each having a success probability p, the
probability of exactly k successes is
This situation is called a Bernoulli Trial. It is a sequence of
n experiments, each with only two possible outcomes, usually named
success with probability p and failure with probability q
= 1p. The probability of success on any experiment is assumed to be
independent of the previous experiments.
See Grinstead and Snell Chapter 3
Grinstead and Snell, Chapter 3 (pdf)
 Wed. Jan. 24

To transfer files between computers, one standard method is to use FTP.
A simple version comes on almost all computers. A slicker windowing
program is WS FTP. Penn students can download it for free:
WS FTP
 Thurs. Jan. 25
 Day of the Week: Given a month, day, and year, how do you
compute the day of the week? Pick a reference day, say Jan. 1, 2001
which was a Monday. Find how many days have elapsed between then and
your target date. Thus, if 14 or 700 days have elapsed, then since
these numbers are divisible by 7, the given date is also a Monday. If
702 days have elapsed it is a Wednesday.
Example: June 1, 2001 We need to observe that June 1 is the
152nd day in 2001 (since 31+28+31+30+31+1 = 152) so it is 1521 = 151
days after Jan. 1. Because 151 = 21*7 + 4, we know the day is 4 days
later, a Friday.
Example: June 1, 2017 We know that 16 full years have
elapsed making 16*365 days, to which we need to add 4 days for the leap
years 2004, 2008, 20012, and 2016. Finally, as above we add 1521=151
to compute that N = 16*365 + 4 + 151 days have elapsed since Jan. 1,
2001. To determine the day of the week we need only the remainder when
dividing N by 7. It is 3 so the day is three days after Monday: a
Thursday.
To check these computations, on a Unix computer the command
cal 2017 gives a calendar for the year 2017, while
cal 6 2017 gives a calendar for June 2017.
Conditional Probabilities
See
The Legacy of the Reverend Bayes and
Grinstead and Snell, Chapter 4 (pdf)
Baysean Inference  Tutorials and Resources
 Tues. Jan. 30
 If you want to post something on the web, where on your computer do you
put it? The answer depends on how the computer was setup.
On mail.sas, eniac.seas and the class computer johnny.sas files you have go
in the directory named html, a subdirectory of your
"home" directory. They can also go in subdirectories of this directory.
Although most computers do not allow you to have programs on the
web, our class does allow this. However, for security, they can only
be in the directory html/cgibin/ (or its subdirectories).
Any programs on the web must allow anyone to execute them. On a
Unix computer this usually means
chmod 755 [program_name]
Programs not intended for the web can be located in any of your
directories. You may wish to make a directory 210 for our class stuff:
mkdir $HOME/210
Although we will primarily use Perl in class, please feel free to use
any other computer language you may prefer. Just don't assume that I
will be able to help you with it.
A simple Perl example. This will also run on mail.sas or eniac.seas,
except that you may need to change the location of perl. [To
find the location just type which perl.]
A
Web form that collects input for
an addition Perl program. This is essentially identical to the previous
line, except using the Web for input.
Another example using the Web for input
A similar script, only using JavaScript instead of Perl. This is
entirely selfcontained. View the Page Source to see the details.
adding,
using JavaScript.
See
Voting on the Web for a sequence of Perl examples I wrote for this class.
See also
Math 210 Bibliography for some useful online Perl references.
 Wed. Jan. 31
 In a Perl program, how do you supply input? In the simple example
perl_example0 the values of the numbers to be added were simply
typed in the program. Here are two more graceful methods:
ask the user:
input: ask
command line
input: program command line
Copy these programs to your own directory and try them. Remember to
change the permissions so these programs are executable:
chmod 755 [name_of_program] .
Some Perl examples:
 test for equality of numbers: if ($x ==3 ) {$y = 2 };
 test for equality of strings:
if ($name eq "Brian") {print "\n\tHi $name\n"; }
 integer division remainder: (19 % 4) gives the remainder
when dividing 19 by 4.
 integer part of a number: int(17.34) gives 17 .
Note int(17.34) gives 17, which may not be what you
want (int always rounds toward 0). Use floor to get
rounding to the next lower integer. Thus floor(17.34) gives
18. Since floor is not a standard part of perl, before using
it you need to have one line with the following:
use POSIX "floor";
 Random numbers. $x = rand(6) assigns $x a random real number
0 <= $x < 6 .
To simulate random heads/tails you can use $toss =
int(rand(2)), where 0 means Heads and 1 Tails.
To simulate a roll of a die, try $roll = int(rand(6)) +1.
Conditional probability (more). A tree diagram is often useful
when analyzing a situation involving conditional probability.
See the discussion in Section 4.1 of
Grinstead and Snell, Chapter 4 (pdf)
 Thursday, Feb. 1
 Computer simulation. Frequently it is helpful to use the computer to
simulate some experiment. Here is a simple one for tossing a coin. We
toss a coin 5000 times and record how many Heads there are.
Coin Toss Simulation
Day of the week. Here is one version. It asks for the date.
Day of Week (perl)
If one cleans up, this appears much shorter (but less clear):
Day of Week
The computation of binomial probabilities (see Jan. 23)
can be painful for n large. Poisson found a useful approximation for
large n and small p, that is, if we let n get very large but keep np
fixed. His approximation is:
In class we used this to compute the answer in an application
where n =12,000 and p = 1/8,000  in which case the approximation
is excellent.
See also Chapter 5 of Grinstead and Snell.
 Tuesday, Feb. 6
 Today we discussed the "Bus Problem" (see the homework) as an
entry to the topic continuous probability distributions. This is
treated in all probability books.
 Wednesday, Feb. 7
 Discussion of some of the homework problems.
Computer graphics using matrices  and an example using Maple:
The Letter F (see also
F page 1 and F page 2).
 Thursday, Feb. 8
 More on continuous probability distributions.
An introduction to
Markov Chains.
Here is a typical (simple) example:
There are two local branches of the Limousine Rental
Company, one at the Airport and one in the City, as well as branches
Elsewhere.
Say every week of the limousines rented from the Airport 25% are returned to
the City and 2% to branches located Elsewhere. Similarly of the
limousines rented from the City 25% are returned to Airport and 2% to
Elsewhere. Finally, say 10% of the limousines rented from Elsewhere are
returned to the Airport and 10% to the City.
If initially there are 35 limousines at the Airport, 35 in the City, and
150 Elsewhere, what is the longterm distribution of the limousines?