Introduction to Statistics

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Introduction to Statistics

• Dr. P Murphy

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Why study Statistics?

• We like to think that we have control over our
  lives.
• But in reality there are many things that are
  outside our control.
• Everyday we are confronted by our own ignorance.
• According to Albert Einstein
• God does not play dice.
• But we all should know better than Prof.
  Einstein.
• The world is governed by Quantum Mechanics where
  Probability reigns supreme.

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Consider a day in the life of an average UCD
student.

• You wake up in the morning and the sunlight hits
  your eyes. Then suddenly without warning the
  world becomes an uncertain place.
• How long will you have to wait for the Number 10
  Bus this morning?
• When it arrives will it be full?
• Will it be out of service?
• Will it be raining while you wait?
• Will you be late for your 9am Maths lecture?

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Probability is the Science of Uncertainty.

• It is used by Physicists to predict the
  behaviour of elementary particles.
• It is used by engineers to build computers.
• It is used by economists to predict the behaviour
  of the economy.
• It is used by stockbrokers to make money on the
  stockmarket.
• It is used by psychologists to determine if you
  should get that job.

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What about Statistics?

• Statistics is the Science of Data.
• The Statistics you have seen before has been
  probably been Descriptive Statistics.
• And Descriptive Statistics made you feel like
  this .

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What is Inferential Statistics?

• It is a discipline that allows us to estimate
  unknown quantities by making some elementary
  measurements.
• Using these estimates we can then
• make Predictions and Forecast the Future

A Crystal Ball
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Chapter 1 Probability
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Consider a Real Problem

• Can you make money playing the Lottery?
• Let us calculate chances of winning.
• To do this we need to learn some basic rules
  about probability.
• These rules are mainly just ways of formalising
  basic common sense .
• Example What are the chances that you get a HEAD
  when you toss a coin?
• Example What are the chances you get a combined
  total of 7 when you roll two dice?

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1.1 Experiments

• An Experiment leads to a single outcome which
  cannot be predicted with certainty.
•  Examples-
• Toss a coin head or tail
•   Roll a die 1, 2, 3, 4, 5, 6
• Take medicine worse, same, better
•     
• Set of all outcomes - Sample Space.
• Toss a coin Sample space h,t
• Roll a die Sample space 1, 2, 3, 4, 5, 6 

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1.2 Probability

• The Probability of an outcome is a number
  between 0 and 1 that measures the likelihood that
  the outcome will occur when the experiment is
  performed. (0impossible, 1certain).
• Probabilities of all sample points must sum to
  1.
•   Long run relative frequency interpretation.
• EXAMPLE Coin tossing experiment
•  P(H)0.5 P(T)0.5

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1.3 Events

• An event is a specific collection of sample
  points.
• The probability of an event A is calculated by
  summing the probabilities of the outcomes in the
  sample space for A.
•  

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1.4 Steps for calculating Probailities

• Define the experiment.
• List the sample points.
• Assign probabilities to the sample points.
• Determine the collection of sample points
  contained in the event of interest.
• Sum the sample point probabilities to get the
  event probability.

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ExampleTHE GAME Of CRAPS

• In Craps one rolls two fair dice.
• What is the probability of the sum of the two
  dice showing 7?

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1.5 Equally likely outcomes

• So the Probability of 7 when rolling two dice is
  1/6
• This example illustrates the following rule
• In a Sample Space S of equally likely outcomes.
  The probability of the event A is given by
• P(A) A / S
• That is the number of outcomes in A divided by
  the total number of events in S.

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1.6 Sets
A compound event is a composition of two or more
other events.  AC The Complement of A is the
event that A does not occur A?B The Union of
two events A and B is the event that occurs if
either A or B or both occur, it consists of all
sample points that belong to A or B or both.  
A?B The Intersection of two events A and B is
the event that occurs if both A and B occur, it
consists of all sample points that belong to both
A and B
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1.7 Basic Probability Rules

• P(Ac)1-P(A)
• P(A?B)P(A)P(B)-P(A?B)
• Mutually Exclusive Events are events which cannot
  occur at the same time.
• P(A?B)0 for Mutually Exclusive Events.

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1.8 Conditional Probability

• P(A B) Probability of A occuring given that B
  has occurred.
• P(A B) P(A?B) / P(B)
• Multiplicative Rule
• P(A?B)
• P(AB)P(B)
• P(BA)P(A)

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1.9 Independent Events

• A and B are independent events if the occurrence
  of one event does not affect the probability of
  the othe event.
• If A and B are independent then
• P(AB)P(A)
• P(BA)P(B)
• P(A?B)P(A)P(B)

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Chapter 1 ProbabilityEXAMPLES
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Probability as a matter of life and death
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Positive Test for Disease

• 1 in every 10000 people in Ireland suffer from
  AIDS
• There is a test for HIV/AIDS which is 95
  accurate.
• You are not feeling well and you go to hospital
  where your Physician tests you.
• He says you are positive for AIDS and tells you
  that you have 18 months to live.
• How should you react?

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Positive Test for Disease

• Let D be the event that you have AIDS
• Let T be the event that you test positive for
  AIDS
• P(D)0.0001
• P(TD)0.95
• P(DT)?

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Positive Test for Disease
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Chapter 1Examples

• Example 1.1
• SA,B,C
• P(A) ½
• P(B) 1/3
• P(C) 1/6
• What is P(A,B)?
• What is P(A,B,C)?
• List all events Q such that
• P(Q) ½.

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Chapter 1Examples

• Example 1.2
• Suppose that a lecturer arrives late to class 10
  of the time, leaves early 20 of the time and
  both arrives late AND leaves early 5 of the
  time.
• On a given day what is the probability that on a
  given day that lecturer will either arrive late
  or leave early?

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Chapter 1Examples

• Example 1.3
• Suppose you are dealt 5 cards from a deck of 52
  playing cards.
• Find the probability of the following events
• 1. All four aces and the king of spades
• 2. All 5 cards are spades
• 3. All 5 cards are different
• 4. A Full House (3 same, 2 same)

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Chapter 1Examples

• Example 1.4
• The Birthday Problem
• Suppose there are N people in a room.
• How large should N be so that there is a more
  than 50 chance that at least two people in the
  room have the same birthday?

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Chapter 1Examples

• Example 1.4
• Children are born equally likely as Boys or Girls
• My brother has two children (not twins)
• One of his children is a boy named Luke
• What is the probability that his other child is a
  girl?

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Example 1.5The Monty Hall Problem

• Game Show
• 3 doors
• 1 Car 2 Goats
• You pick a door - e.g. 1
• Host knows whats behind all the doors and he
  opens another door, say 3, and shows you a goat
• He then asks if you want to stick with your
  original choice 1, or change to door 2?

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Ask Marilyn.Parade Magazine Sept 9 1990

• Marilyn vos Savant
• Guinness Book of Records -Highest IQ
• Yes you should switch. The first door has a 1/3
  chance of winning while the second has a 2/3
  chance of winning.
• Ph.D.s - Now two doors, 1 goat 1 car so chances
  of winning are 1/2 for door 1 and 1/2 for door
  2.
• You are the goat - Western State University.

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Whos right?

• At the start, the sample space is
• CGG, GCG, GGC
• Pick a door e.g. 1
• 1 in 3 chance of winning
• Host shows you a goat so now
• CGG, GCG, GGC
• So Marilyn was right, you should switch.

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Not convinced?

• Imagine a game with 100 doors.
• 1 F430 Ferrari, 99 Goats.
• You pick a door.
• Host opens 98 of the 99 other doors.
• Do you stick with your original choice? Prob
  1/100
• Or move to the unopened door.
• Prob 99/100

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Boys, Girls and Monty Hall

• Sample Space ( listing oldest child first)
• GG, BG, GB, BB
• Equally likely events
• One child is a boy
• GG is impossible
• BG, GB, BB gt
• P(OC G) 2/3
• Luke is 6 months old.
• GB, BB gt P(OC G) 1/2

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Odd Socks
It is winter and the ESB are on strike. This
morning when you woke up it was dark. In your
sock drawer there was one pair of two black socks
and one odd brown one. Are you more or less
likely to be wearing matching socks today?
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EXAMS
Campus Female Pass Rate Male Pass Rate
Belfield 40 33
ET/ Carysfort etc. 75 71
Seeing this evidence amale student takes UCD to
court saying there is discimination against male
students. UCD gathers all its exam information
together and reports the following.
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EXAM Pass Rates
Overall Female pass rate is 56 Overall Male pass
rate is 60 HOW CAN THIS BE? Clearly UCD are
LYING !
Campus Female Pass Rate Male Pass Rate
Belfield 40 33
ET/ Carysfort etc. 75 71
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Simpsons Paradox
Overall Female pass rate is 56 Overall Male pass
rate is 60
Campus Female Pass Rate Male Pass Rate
Belfield 40 20/50 33 10/30
ET/ Carysfort etc. 30/40 75 50/70 71
50/90 56 60/100 60
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Hit and RUN
Once upon a time in Hicksville, USA there was a
night-time hit and run accident involving a taxi.
There are two taxi companies in Hicksville,
Green and Blue. 85 of taxis are Green and 15
are Blue. A witness identified the taxi as being
Blue. In the subsequent court case the judge
ordered that the witnesss observation under the
conditions that prevailed that night be tested.
The witness correctly identified each colour of
taxi 80 of the time.
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Hit and RUN
What is the probability that it was indeed a blue
taxi that was involved in the accident?
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DNA
You are holiday in Belfast and an explosion
destroys the Odessey arena. You are seen
running from the explosion and are arrested. You
are subsequently charged with being a member of a
prescribed paramilitary organisation and with
causing the explosion. In court you protest your
innocence. However the PSNI have DNA evidence
they claim links you to the crime.
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DNA
Their forensic scientist delivers the following
vital evidence. The forensic scientist indicates
that DNA found on the bomb matches your DNA.
Your lawyer at first disputes this evidence and
hires an independent scientist. However the
second forensic scientist also says that the DNA
matches yours and that there is a 1 in 500
million probability of the match.
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DNA
What do you do? It appears as if you are going to
spend the rest of your days in jail.
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The National Lottery
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I lied, cheated and stole to become a
millionaire. Now anybody at all can win the
lottery and become a millionaire
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GAME 1 LOTTO 6/42

• What are the chance of winning with one selection
  of 6 numbers?
• Matches Chances of Winning
  
• 6 1 in 5,245,786
• 5 1 in 24,286
• 4 1 in 555

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GAME 1 LOTTO 6/42

• Expected Winnings
• Only consider Jackpot
• 1 Euro get 1 play
• E(win) Jackpot(1/5,245,786) 1Euro(5,245,785/5,
  245,786)
• E(win)
• Jackpot0.000000191- 0.999999809
• If only one jackpot winner then
• Positive E(win) if
• Jackpot gt5,245,785

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LOTTO 6/42

• The average time to win each of the prizes is
  given by
• Match 3 with Bonus 2 Years, 6 Weeks
• Match 4 2 Years, 8 Months
• Match 5 116 Years, 9 Months
• Match 5 with Bonus 4323 Years, 5 Months
• Share in Jackpot 25,220 Years

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Tossing a fair coin
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Tossing a coin!

• You are joking!
• That is boring no question about it!
• 1957 Second edition of William Fellers Textbook
  includes a chapter on coin-tossing.
• Introduction The results concerning
  coin-tossing show that widely held beliefs
  are fallacious. These results are so amazing and
  so at variance with common intuition that even
  sophisticated colleagues doubted that coins
  actually misbehave as theory predicts.

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Tossing a coin!

• Toss a coin 2N times.
• Law of Averages
• As N increases the chances that there are equal
  numbers of heads and tails among the 2N tosses
  increases.
• Lim N-gt? P( H T ) 1.
• In the limit as N tends to infinity the
  probability of matching numbers of heads and
  tails approaches 1.

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Rosencrantzand Guildenstern are Dead
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Prob of equal numbers of H and T
of tosses 2 4 6 8 10
½ 3/8 5/16 35/128 63/256
Prob 0.5 0.375 0.3125 0.273 0.246