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Week 10 - Monday
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What did we talk about last time?
Combinations
Binomial theorem
A bundle of 120 wires has been laid underground
between two telephone exchanges 10 miles apart
 Unfortunately, it was discovered that the individual
wires are not labeled
 Visually, there is no way of knowing which wire is
which, making connections at either end impossible
 Your job is to label the wires at both ends
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 Walking is your only transportation
 You have a battery and a light bulb to test continuity
 You have tape and a pen for labeling the wires
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What is the shortest distance in miles you will need to
walk to correctly identify and label each wire?
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Consider the numbers 1 through 99,999 in
their ordinary decimal representations.
How many contain exactly one of each of the
digits 2, 3, 4, and 5?
 For example, 53,142 counts but 53,541 does not
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On an 8 × 8 chessboard, a rook is allowed to
move any number of squares either
horizontally or vertically.
How many different paths can a rook follow
from the bottom-left square of the board to
the top-right square of the board if all moves
are to the right or upward?
Hint: Think of representing each move as an
R or a U
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A bakery produces six different kinds of
pastry, one of which is eclairs. Assume there
are at least 20 pastries of each kind.
 How many different selections of twenty pastries
are there?
 How many different selections of twenty pastries
are there if at least three must be eclairs?
 How many different selections of twenty pastries
contain at most two eclairs?
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How many different solutions are there to the
following equation, assuming that each xi is a
nonnegative integer?
x1 + x2 + x3 = 20
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What if each xi is a positive integer?
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a + b is called a binomial
Using combinations (or Pascal's Triangle) it is
easy to compute (a + b)n
n
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 n k k
n
(a  b)    a b
k 0  k 
n
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Compute (2x + 3)7 using the binomial theorem
Student Lecture
If n pigeons fly into m pigeonholes, where n > m,
then there is at least one pigeonhole with two or
more pigeons in it
 More formally, if a function has a larger domain
than co-domain, it cannot be one-to-one
 We cannot say exactly how many pigeons are in
any given holes
 Some holes may be empty
 But, at least one hole will have at least two
pigeons
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A sock drawer has white socks, black socks,
and red argyle socks, all mixed together,
What is the smallest number of socks you
need to pull out to be guaranteed a matching
pair?
Let A = {1, 2, 3, 4, 5, 6, 7, 8}
If you select five distinct elements from A,
must it be the case that some pair of integers
from the five you selected will sum to 9?
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If n pigeons fly into m pigeonholes, and for
some positive integer k, n > km, then at least
one pigeonhole contains k + 1 or more
pigeons in it
Example:
 In a group of 85 people, at least 4 must have the
same last initial
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Let A and B be events in the sample space S
 0 ≤ P(A) ≤ 1
 P() = 0 and P(S) = 1
 If A  B = , then P(A  B) = P(A) + P(B)
 It is clear then that P(Ac) = 1 – P(A)
 More generally, P(A  B) = P(A) + P(B) – P(A  B)
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All of these axioms can be derived from set
theory and the definition of probability
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What is the probability that a card drawn
randomly from an Anglo-American 52 card
deck is a face card (jack, queen, or king) or is
red (hearts or diamonds)?
Hint:
 Compute the probability that it is a face card
 Compute the probability that it is red
 Compute the probability that it is both
Expected value is one of the most important
concepts in probability, especially if you want to
gamble
 The expected value is simply the sum of all
events, weighted by their probabilities
 If you have n outcomes with real number values
a1, a2, a3, … an, each of which has probability p1,
p2, p3, … pn, then the expected value is:
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n
a p
k 1
k
k
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A normal American roulette wheel has 38 numbers: 1
through 36, 0, and 00
18 numbers are red, 18 numbers are black, and 0 and 00 are
green
The best strategy you can have is always betting on black (or
red)
If you bet $1 on black and win, you get $1, but you lose your
dollar if it lands red
What is the expected value of a bet?
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Given that some event A has happened, the
probability that some event B will happen is
called conditional probability
This probability is:
P ( A  B)
P(B | A) 
P( A)
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Given two, fair, 6-sided dice, what is the
probability that the sum of the numbers they
show when rolled is 8, given that both of the
numbers are even?
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Let sample space S be a union of mutually
disjoint events B1, B2, B3, … Bn
Let A be an event in S
Let A and B1 through Bn have non-zero
probabilities
For Bk where 1 ≤ k ≤ n
P( A | Bk )  P(Bk )
P(Bk | A) 
P( A | B1 )  P(B1 )  P( A | B2 )  P(B2 )  ...  P( A | Bn )  P(Bn )
Bayes' theorem is often used to evaluate tests that can
have false positives and false negatives
 Consider a test for a disease that 1 in 5000 people have
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 The false positive rate is 3%
 The false negative rate is 1%
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What's the probability that a person who tests positive for
the disease has the disease?
Let A be the event that the person tests positively for the
disease
Let B1 be the event that the person actually has the
disease
Let B2 be the event that the person does not have the
disease
Apply Bayes' theorem
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If events A and B are events in a sample space S ,
then these events are independent if and only if
P(A  B) = P(A)∙P(B)
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This should be clear from conditional probability
If A and B are independent, then P(B|A) = P(B)
P ( A  B)
P(B | A)  P(B) 
P( A)
P( A)  P(B)  P( A  B)
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Finish probability
Graph basics
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Finish reading Chapter 9
Work on Homework 8
 Due next Friday
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Start reading Chapter 10