Math 210G.M01, Fall 2013
Lecture 6: Combinatorial
aspects of probability
Combinatorics
Probability versus Likelihood
• Cardano’s classical definition of probability:
If the total number of possible outcomes,
all equally likely, associated with some
actions is n and if m of those n result in the
occurrence of some given event, then the
probability of that event is m/n.
• Question: how do we know if certain
events are equally likely?
Is the coin fair?
• A fair coin should land on Heads 50% of
the time.
• A fair coin tossed once will either land on
heads or on tails.
• From one trial it is impossible to
determine if the coin is fair.
• A fair coin, tossed 100 times, might land
on heads 60 times, but how likely is that to
happen?
Likelihood
• Ocham’s razor says that we should use the simplest
model that fits the data.
• If a coin comes up 60 times in 100 trials, then the
simplest hypothesis is that the coin is biased
towards landing on heads.
• However, suppose the coin looks exactly like other
coins that, in our experience, are fair coins.
• Now we have two sets of data: the results of the
coin flips, and how the coin looks.
• What is the simplest model that fits the data?
What’s the moral lesson
here?
• In games of chance (poker, craps, etc) the
problem is to apply combinatorial
calculations to determine probabilities of
given events
• In other parts of life, the bigger problem is
to try to find the most likely model that
explains the data.
• Often personal experience misleads us: I
got a C in logic. My friend got a C in logic. It’s
impossible to get an A in logic.
Question to Ponder for Later
• Postulate: a person should vote to elect the
candidate under whose leadership the
individual is most likely to be better off.
• Question: Will the most number of people
actually be better off if the person who meets
this criterion for the most number of people
is elected?
If you flip a penny 100 times, how
many heads and tales do you
expect?
Coin flipping sites
• Random.org: Click on games and coin
flipper. Does not tally heads.
• Whidbey simulated coin tosser : tallies
heads and tails
• Mathsonline: shows cumulative histogram
but only 10 coins allowed. Seems biased.
• BTWaters: effective for looking at results
of large numbers of coin flips
• Ken White’s coin flipping page: shows
results. Posts historical data.
Binomial distribution:
• Independent events: the outcome (H,T) of the
second coin does not depend on the outcome
of the first.
• Typical sequence of result of 10 flips:
• HTTHTTTHTH
• Given N fair coins, the probability of any
given outcome sequence is
(1/2)*(1/2)*…*(1/2)=1/2^N
• The probability of HTTHTTTHTH is
(1/2)^10=1/1024
• This is the same as the probability of
• HHHHHHHHHH
• What does typical mean?
• Order matters
What if order doesn’t matter?
•
•
•
•
Two coins: the possible outcomes are:
1) TT 2) TH 3) HT 4) HH
Each with probability ¼
The probability of one head and one tail is
equal to ½ since it can happen two
different ways.
Clicker question 1
• If you flip three coins, what is the
probability that they all come up heads?
A)½
B)¼
C)1/8
D)3/8
Clicker question 2
• If you flip three coins, what is the
probability that exactly TWO of them
come up heads?
A)½
B)¼
C)3/8
D)¾
E)None of these
Clicker question 3
• If you flip three coins, what is the
probability that exactly ONE of them
come up heads?
A)½
B)¼
C)3/8
D)¾
E)None of these
Choosing subsets
• A set of N elements has 2^N subsets if we
include the empty set and the whole set.
• Think of the set a set of N coins and the
“chosen” subset of the ones that will be
heads.
• Binomial coefficients
Factorials
•
•
•
•
•
•
•
•
0!=1
1!=1
2!=2
3!=3*2=6
4!=4*3!=4*6=24
5!=5*24=120
6!=6*120=720
7*=7*720=5040 etc
Stirling’s approximation
(Euler’s number)
N choose k
• N choose K equals…
• N-1 choose K plus N-1 choose K-1
• The number of distinct ways in which to
choose K elements from a set of N
elements
• Fix one element. If it is not chosen, all K
must be from the remaining N-1. If it is
chosen, the remaining K-1 must come out
of the remaining N-1
N choose k
Group assignment
• Please compute the next row of the table
above. Turn in a sheet of paper with the
title “The number of ways of choosing N
elements from a set of 11 elements” the
numbers from left to right, and the names
of those in your group, and today’s date.
You will be given 5 minutes for this. You
are allowed to use your cell phone, but
must turn off your phone when you are
finished.
Clicker question
• How many ways are there to choose one
element from a set of 11 elements?
• A) 1
• B) 5
• C) 6
• D 11
Clicker question
• How many ways are there to choose 5
distinct elements from a set of 11 distinct
elements?
N Choose K revisited
•The number on the left is the same as “n
choose k”
•This formula is useful for computing the
binomial coefficient n choose k when n is
large.
Example
• 52 choose 5: number of to choose a 5 card
poker hand from a set of 52 poker cards.
• But
=2,598,960
Quetelet’s graph
Plotting Pascal’s triangle
• The web page:
http://www.ams.org/samplings/featurecolumn/fcarc-normal shows plots of the
numbers in several rows of Pascal’s
triangle.
• For large row numbers, the row plots look
like a bell-shaped curve
Normal approximation to
binomial
Normal approximations,
N=10, 100, 1000
Red curves give idealized normal approximation for a fair coin flipped N times.
Blue histograms give probabilities of outcomes for biased N flips of a coin that has
a 70% chance of landing on heads.
Overlapping distributions
• The distributions illustrate different probabilities. When
two distributions have a lot of overlap, it is not clear
whether an event should be associated with one
distribution as opposed to the other.
• Conversely, when two distributions are separated, the
chance that an event will mistakenly be associated with
the wrong one is very small.
• For the fair coin problem, the distributions become more
separated as the number of trials is increased.
• We will see later that normal curves give a means to
calculate overlaps and associate probabilities to them.
Likelihood
• In statistics, a likelihood function is a function of
the parameters of a statistical model, defined as
follows: the likelihood of a set of parameter values
given some observed outcomes is equal to the
probability of those observed outcomes given
those parameter values. Likelihood is a function
of the data.
Law of Large numbers
• The law of large numbers states that if X1,
X2 ,…, Xn are independent samples of a
random variable then the average value
approaches the expected value as the
number of trials tends to infinity.
• In the case of a fair coin, counting 1 for
heads and 0 for tails, the average value
after a large number of trials should
approach ½.
Gambler’s ruin
• The law of large numbers is sometimes
misinterpreted as suggesting that if a coin
comes up tails (or similar unfavorable
event) occurs several consecutive times
then the coin is more likely to come up
heads the next time. This contradicts the
hypothesis of independence.
• The full central limit theorem indicates that
as the sample size N increases, the
distribution of the sample average of these
binomial “random variables” approaches the
normal distribution.
• The central limit theorem was postulated by
Abraham de Moivre who, in a remarkable
article published in 1733, used the normal
distribution to approximate the distribution
of the number of heads resulting from many
tosses of a fair coin.
• This finding was ahead of its time, and
nearly forgotten until Pierre-Simon Laplace
rescued it from obscurity in his monumental
work Théorie Analytique des Probabilités,
published in 1812. Laplace expanded De
Moivre's finding by approximating the
binomial distribution with the normal
distribution.
• As with De Moivre, Laplace's finding
received little attention in his own time. It
was not until the nineteenth century was
at an end that the importance of the
central limit theorem was discerned,
when, in 1901, Russian mathematician
Aleksandr Lyapunov defined it in general
terms and proved precisely how it worked
mathematically. Nowadays, the central
limit theorem is considered to be the
unofficial sovereign of probability theory.
Galton board illustrated
Second application: card games
• 5 card poker hands
• The number ways of choosing 5 cards
from a set of 52 cards is “52 choose 5”
• =2,598,960
Probabilities as proportions
• Number of favorable outcomes divided by
total number of possible outcomes
• Chance of 4 of a kind: 13*48 out of
2,598,960
• 0.00024
• 240 out of a million
Possible poker hands
Straight flush
Four of a kind
Full house
Flush (nonconsecutive)
Straight (mixed)
Three of a kind
Two pairs
One pair
No pairs
Total
40
624
3,744
5,108
10,200
54,912
123,552
1,098,240
1,302,540
2,598,960
How to figure…
• The number of ways to get a straight…
• Starting rank: 10 possible
A,K,Q,J,10,9,8,7,6,5
• Number of ways from a given starting
rank: 4x4x4x4x4 = 1024
• Total: 10,240
• Subtract straight flushes: 10,200
How to figure…
The number of ways to get 3 of a kind…
Rank: 13 possible
Number of a given rank: “4 choose 3” = 4
Number of possibilities of remaining two
cards that do not give a pair: 48x44/2
• Total: 13x4x48x22=54912
•
•
•
•
Problem
• Show how to determine the number of
ways in which to get a poker hand
containing exactly a pair.
Clicker question
• Which 5 card poker hand has greater
odds?
A) Full house
B) straight
C) flush
D) Two pair
Clicker question
• The number of distinct poker hands that
have two pair but not three pair or higher:
A) 127,920
B) 123,552
C) 1,098,240
D) 247,105
Group work problem
• Three card guts is a poker game that
involves three cards. Straights and pairs
are not counted. The best possible hand is
three of a kind.
• To do: figure out with your mates how
many three card guts hands there are, and
how many of them have a pair or better.
• Turn in: your solutions, today’s date and
names of those in your group.
Exercise 0
• A fair coin is flipped 10 times.
• What is the probability that it will come
up heads 5 times?
• What is the probability that it will come
up heads 6 time?
• What is the probability that it will come
up heads 7 times?
• What is the probability that it will come
up heads 8 times?
Exercise 1
A fair coin is tossed three times. What is the
probability that it will land on heads one
or two times? What is the probability that
it will land on heads two or three times?
A fair coin is flipped 10 times. What is the
probability that it will land on heads
exactly 5 times? What is the probability
that it will land on heads 5 or more times?
What is the probability that it will land on
heads fewer than five times?
Exercise 2
• Compute the number 10 choose 2 and 10
choose 8.
• Compute the numbers 10 choose 5, 10
choose 4 and 10 choose 6
• Compute the number 52 choose 47 and 52
choose 5
Exercise 2.5
• Suppose that the probability of a child being born a boy
is the same as the probability of a child being born a girl.
• In a family of ten children, what is the probability that
exactly six of the children will be boys? What is the
probability that exactly six will be girls?
• What is the probability that at least six of the children
will be girls?
• What is the probability that they will all be girls?
Exercise 3
• 4 kids want to play a game of two on two
basketball. How many ways are there to
divide the four players into two teams of
two players each?
• 10 kids want to play a game of 5 on 5.
How many different ways are there of
dividing the 10 players into two teams of 5
each?
Exercise 4
• How many distinct three card guts hands
are there?
• How many three card guts hands contain
three of a kind?
• How many three card guts hands contain
a pair but not three of a kind?
• How many three card guts hands to not
contain any pairs?
Exercise 5
• Explain the difference between a likelihood
and a probability