Math 210G.M03, Spring 2014

Lecture 6: Combinatorial aspects of probability

Part I: Binomial distributions

• 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. • 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

• The probability of HTTHTTTHTH is

• 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:

• 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) ½B) ¼C) 1/8D) 3/8

Clicker question 2

• If you flip three coins, what is the probability that exactly TWO of them come up heads?

A) ½B) ¼C) 3/8D)¾E) None of these

Clicker question 3

• If you flip three coins, what is the probability that exactly ONE of them come up heads?

A) ½B) ¼C) 3/8D)¾E) 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

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/feature-column/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 40

Four of a kind 624

Full house 3,744

Flush (nonconsecutive)

5,108

Straight (mixed) 10,200

Three of a kind 54,912

Two pairs 123,552

One pair 1,098,240

No pairs 1,302,540

Total 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 houseB) straightC) flushD)Two pair

Clicker question

• The number of distinct poker hands that have two pair but not three pair or higher:

A) 127,920B) 123,552C) 1,098,240D) 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 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 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 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 do not contain any pairs?

Exercise 5

• Explain the difference between a likelihood and a probability