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Math 210G.M03, SPring 2017
Lecture 6: Combinatorial
aspects of probability
Activity
• Roll a die 10 times.
• Record the number facing up on each roll.
• How many times was the number rolled
an even number?
Clicker question
• How many times was the number that
you rolled an even number?
• Choices: 0, 1, 2, 3,…,10
Part I: Binomial distributions
An unfair coin.
• A fair coin is one that comes up heads with a
probability equal to ½ . How can we tell that
a coin is fair?
• Suppose that a coin is slightly unfair. For
example, the probability it comes up heads is
49/100. How can we tell that the coin is
slightly unfair?
• These are not meaningless questions: they
apply to a lot of questions in which precise
likelihoods are important to know.
Roulette
Odds of all black or red
All numbers 0-36
18 of them red, 18 black and one green
Odds of all black: 18/37
Casino’s odds: 19/37
Casino’s advantage: 19:18. On average, for every 18
times it lands on black, it will not land on black 19 times.
Better than a 5 percent advantage.
• You would need to toss a coin a lot of times to determine
it is unfair, if the odds of heads are 19/37.
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• Binomial distributions refer to
distributions of probabilities of outcomes
for a sum of random variables each having
one of two possible values or outcomes.
• Example: flipping one coin: H or T
• Count 1 for H and 0 for T.
• Example: flipping N coins and adding up
the number of H.
• This sum can take any value between 0
and N. We are interested in the
probability for each possible value.
• Likelihood: how likely is it that a coin,
flipped N times, and coming up heads M
times, is a fair coin?
• Similar questions arise in real life:
• How likely is it that a coin is fair…
• … or that a defendant is guilty, given the
evidence presented in the trial
• or that a drug is safe and effective, given
results of clinical trials
• Or that some law discriminates unfairly
against some segment of the population,
given pertinent data
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.
• How do we know if certain events are
equally likely?
• In the case of a coin this can lead to
circular reasoning.
Is the coin fair?
• A fair coin should land on Heads 50% of
the time…in the long run.
• 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 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.
• When do we have enough information?
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.
• BTWaters: effective for looking at results
of large numbers of coin flips. Posts
historical data
• Interactive coin tosser. Shows sequence of
T/H
What are reasonable
numbers of heads and tails
to expect?
• Example: In the 2000 general election in
the state of Florida, Gore obtained
2,912,253 and Bush obtained 2,912,790
votes. Total B&G votes: 5825043.
• Difference: 537
• Exercise: flip a fair coin 5,825,043 times.
How many times does it come up heads>
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
• The probability of HTTHTTTHTH is
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This is the same as the probability of
HHHHHHHHHH
What does typical mean?
Order matters
What if order doesn’t matter?
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Two coins: the possible outcomes are:
TT or TH or HT or 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
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
Name_____________
Math 210, March 1, 2016
Name_____________
Fill in the next row of Pascal’s triangle
Name_____________
Name______________
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
Part II: Likelihood revisited
Histograms
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.
BinomialNormal
approximation
• Flipping a coin n^2 times is the same as
flipping n coins n times.
• Let
be the proportion of heads: the
number of heads divided by n.
• The expected value of X_n is the same as
the expected value of X_{n^2}
• But how does the variance of X_n depend
on n?
• Answer:
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
Part III: More fun and games
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
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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
Name_____________
Math 210, March 3, 2016
Name_____________
Computing three card guts hands
Name_____________
Name______________
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•
Three card guts is a poker game that involves three cards. Straights and
flushes 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, todays date and names of those in your group.
Number of possible hands is 52 choose 3. Compute this.
Number of possible pairs: first compute the number of different ways to get
two different ranks, thirteen choose two. Then compute the number of ways
to get a pair in one rank and to get a card of the other rank. Multiply
these.Then multiply by two since the pair can be at the higher rank or at the
lower rank.
Three card guts
•
•
Number of possible hands is 52 choose 3. Compute this.
Number of possible pairs: first compute the number of different ways to get
two different ranks, thirteen choose two. Then compute the number of ways
to get a pair in one rank and to get a card of the other rank. Multiply
these.Then multiply by two since the pair can be at the higher rank or at the
lower rank.
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 0: solution
• A fair coin is flipped 10 times. There are 2^10=1024 possible
outcomes.
• The number of outcomes that result in 5 heads is the same as the
number of ways of choosing 5 object (the coins that come up heads)
from 10 objects, 10 choose 5 is 252. So the probability is p=252/1024
=0.2461
• 10 choose 6 is 210. p=210/1024= 0.2051
• 10 choose 7 = 120 so p=120/1024=0.1172
• 10 choose 8 = 45 so p=45/1024=0.0439
Exercise 1
• A fair coin is flipped 100 times. What is
the probability that the coin lands on
heads exactly 50 times? What is the
probability that the coin lands on heads 51
or more times? What is the probability
that the coin lands on heads 49 or fewer
times?
Exercise 1solution
• A fair coin is flipped 100 times. What is the probability that the coin
lands on heads exactly 50 times?
• What is the probability that the coin lands on heads 51 or more
times? What is the probability that the coin lands on heads 49 or
fewer times?
• The two cases are symmetric (landing on heads 51 or more times is
the same as landing on tails 49 or fewer times and has equal
probability of landing on heads 49 or few times. Therefore, the
probability of landing on heads 51 or more times is 1 minus the
probability of landing on heads 50 times, divided by two, or
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 solution
• Compute the number 10 choose 2 and 10 choose 8.
• Similarly,
• Compute the numbers 10 choose 5, 10 choose 4 and 10 choose 6
• Compute the number 52 choose 47 and 52 choose 5
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 3 solution
• The number of ways to choose a team with two players is 4 choose 2
which is 6. However, once one team of two is chosen, the other team
is also set so half of these 4 choose 32 choices are the other team. So
there are actually 6/2=3 ways.
• The number of ways to choose of ateam of 5 players from 10 kids is
10 choose 5 which is 242. However, once one team is chosen, the
other team is automatically set, so half of these choices correspond
to the other team. Hence there are actually 242/2=121 ways.
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 do not
contain any pairs?
Exercise 4 solution
• The number of distinct three car guts hands is 52 choose 3 which is
52x51x50/3x2x1=22,100
• In a three card hand, there are 4 choose 3=4 ways to get three of a
given rank, and there are 13 ranks so there are 4x13=52 ways to get
three of a kind.
• There are 13 choose 2 =78 ways to get two ranks. There are 4 choose
2 = 6 ways to get a pair of a given rank and there are 4 ways to get a
card of a second rank. We have to consider that the pair can be at
the higher or lower rank so there are
• 78x6x4x2=3744 such hands
• How many three card guts hands do not contain any pairs? Subtract
22100-52-3744=18304
Exercise 5
• Explain the difference between a likelihood
and a probability
Exercise 5 solution
• Explain the difference between a likelihood and a probability
• Probability refers to the probability of an event and assumes that
certain outcomes of an event are equally likely, for example, if a coin is
assumed to be fair then its probability of landing on heads or tails is
one half.
• The term likelihood refers technically to values of parameters. For
example, suppose that I coin a priori has an unknown probability of
landing on heads. One then estimates the likelihood that it is a fair coin
(or has another probability of landing on heads) from data such as
tossing it a given number of times and counting the frequency that it
lands on heads.