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## Introduction to Statistics

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**Introduction to Statistics**Dr. P Murphy**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.**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?**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.**What aboutStatistics?**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 ….**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**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?**1.1 Experiments**AnExperiment 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}**1.2 Probability**The Probability of an outcome is anumber between 0 and 1 that measures the likelihood that the outcome will occur when the experiment is performed. (0=impossible, 1=certain). 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**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.**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.**Example:THE GAME Of CRAPS**In Craps one rolls two fair dice. What is the probability of the sum of the two dice showing 7?**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.**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 AB :The Unionof 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. AB:TheIntersectionof 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**1.7 Basic Probability Rules**P(Ac)=1-P(A) P(AB)=P(A)+P(B)-P(AB) Mutually Exclusive Events are events which cannot occur at the same time. P(AB)=0 for Mutually Exclusive Events.**1.8 Conditional Probability**P(A | B) ~ Probability of A occuring given that B has occurred. P(A | B) = P(AB) / P(B) Multiplicative Rule: P(AB) = P(A|B)P(B) = P(B|A)P(A)**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(A|B)=P(A) P(B|A)=P(B) P(AB)=P(A)P(B)**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?**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(T|D)=0.95 • P(D|T)=?**Chapter 1Examples**Example 1.1 S={A,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.**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?**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)**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?**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?**Example 1.5The Monty Hall Problem**• Game Show • 3 doors • 1 Car & 2 Goats • You pick a door - e.g. #1 • Host knows what’s 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?**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.**Who’s 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.**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**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} => • P(OC = G) = 2/3 • Luke is 6 months old. • {GB, BB} => P(OC = G) = 1/2**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?**EXAMS**Seeing this evidence amale student takes UCD to court saying there is discimination against male students. UCD gathers all it’s exam information together and reports the following.**Overall Female pass rate is 56%**Overall Male pass rate is 60% HOW CAN THIS BE? Clearly UCD areLYING ! EXAM Pass Rates**Overall Female pass rate is 56%**Overall Male pass rate is 60% Simpson’s Paradox**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 witness’s observation under the conditions that prevailed that night be tested. The witness correctly identified each colour of taxi 80% of the time. Hit and RUN**What is the probability that it was indeed a blue taxi that**was involved in the accident? Hit and RUN**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. 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. DNA**What do you do?**It appears as if you are going to spend the rest of your days in jail. DNA**“I lied, cheated and stole to become a millionaire. Now**anybody at all can win the lottery and become a millionaire”**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**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)= Jackpot*0.000000191-0.999999809 If only one jackpot winner then: Positive E(win) if Jackpot >5,245,785**LOTTO 6/42**• The average time to win each of the prizes is given by: • Match 3 with Bonus2 Years, 6 Weeks • Match 4 2 Years, 8 Months • Match 5 116 Years, 9 Months • Match 5 with Bonus4323 Years, 5 Months • Share in Jackpot 25,220 Years