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STAT 110 - Section 5 Lecture 20

STAT 110 - Section 5 Lecture 20. Professor Hao Wang University of South Carolina Spring 2012. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. Chapter 17 – Thinking About Chance. What’s the chance of getting killed by lightning?

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STAT 110 - Section 5 Lecture 20

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  1. STAT 110 - Section 5 Lecture 20 Professor Hao Wang University of South Carolina Spring 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  2. Chapter 17 – Thinking About Chance What’s the chance of getting killed by lightning? How could we come up with a number? US population = about 310 million An average of 40 people per year are killed by lightning. Chance of being killed by lightning:40 / 310,000,000 = 0.000013%

  3. Thinking About Chance What does a baseball player’s batting average really mean? Let’s say your favorite player is batting 300. If a player gets 3 hits out of 10 at-bats, he has a batting average of 3 / 10 = 30% For baseball stats, we multiply by 10. So, we have player with a batting average of 300. So, when our player steps up to the plate, he has only a 30% chance of getting a hit.

  4. Randomness and Probability • Random in statistics does not mean “haphazard”. • In statistics, random describes a kind of order that emerges only in the long run. • Probability describes the long-term regularity of events. • Probability describes what happens in very many trials.

  5. The Idea of Probability • Why is random good? Why do we use random samples and randomized experiments? • Why is tossing a coin before a football game reasonable? Is it random? • Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.

  6. Randomness and Probability random – when individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions probability – a number between 0 and 1 that describes the proportion of times an outcome would occur in a very long series of repetitions

  7. Probability • probability = 0  the outcome never occurs • probability = 1  the outcome happens on every repetition • probability = ½  the outcome happens half the time in a very long series of trials • Probability gives us a language to describe the long-term regularity of random behavior.

  8. Myths About Chance • The myth of short-run regularity. • The idea of probability is that randomness is regular in the long run. • Our intuition tells us that randomness should also be regular in the short run. • When regularity in the short run is absent, we look for some explanation other than chance variation.

  9. Example What looks random? Toss a coin six times. Which of these outcomes is more probable? HTHTTH TTTHHH

  10. Example • In basketball, what’s a “hot hand”? • If a player has a hot hand, is he/she more likely to make the next shot? • Do players perform consistently or in streaks? • Runs of hits and misses are more common than our intuition expects. • The evidence seems to be that if a player makes • half his/her shots in the long run, hits and misses • behave like a tossed coin.

  11. Myths About Chance 2. The myth of the surprise meeting. • When something unusual happens, we look back and say, “Wow, what were the chances?” • Say you’re spending the summer in London. You’re at the Tower of London, and you run into an acquaintance from college. • What are the chances?

  12. Example • Cancer accounts for more than 23% of all deaths in the US. • Of 250 million people, this is about 57.5 million people. • How many of those people live in the sample neighborhood? • There are bound to be clusters of cancer cases simply by chance.

  13. Myths About Chance 3. The myth of the law of averages. • Believers in the law of averages think that future outcomes must make up for an imbalance. • For example, if you toss a coin six times and get TTTTTT, the law of averages believers will tell you the next toss will be a H, just so it evens out. • Coins and dice have no memories!

  14. Example • A couple gets married and decides to start a family. • They decide to have four children. • All four are girls. Wanting a boy, they try again. • What happens? • For this couple, having children is like tossing a coin. Eight girls in a row is highly unlikely, but once seven girls have been born, it’s not at all unlikely that the next child will be a girl.

  15. Personal Probability personal probability – an outcome is a number between 0 and 1 that expresses an individual’s judgment of how likely the outcome is • Personal probabilities are not limited to repeatable settings. • They’re useful because we base decisions on them.

  16. Personal Probability • Personal probability expresses individual opinion. • It can’t be said to be right or wrong. • It is NOT based on many repetitions. • There is no reason why a person’s degree of confidence in the outcome of one try must agree with the results of many tries. • What’s the probability that USC will win the football game this weekend?

  17. Probability and Risk • We have two types of probability. • One looks at “personal judgment of how likely”. • The other looks at “what happens in many repetitions”. • Experts tend to look at “what happens in many repetitions”, and the public looks to “personal judgment”. • So, which should we use? Say we want to know • what’s considered risky?

  18. Probability and Risk http://www.idsnews.com/news/story.aspx?id=77905 http://www.livescience.com/environment/050106_odds_of_dying.html http://reason.com/archives/2006/08/11/dont-be-terrorized http://www.independent.co.uk/news/world/americas/us-living-in-fear-over-summer-of-child-abductions-649895.html

  19. http://finance.yahoo.com/news/never-safer-fly-deaths-record-163444463.htmlhttp://finance.yahoo.com/news/never-safer-fly-deaths-record-163444463.html • That's two deaths for every 100 million passengers on commercial flights, according to an Associated Press analysis of government accident data. • You are more likely to die driving to the airport than flying across the country. There are more than 30,000 motor-vehicle deaths each year, a mortality rate eight times greater than that in planes.

  20. Probability and Risk • Why is there such a difference between what we consider risky and what the experts consider risky? • We feel safer when a risk seems under our control than when we can’t control it. • It’s hard to comprehend small probabilities. • The probabilities for some risks are estimated by experts from complicated statistical studies.

  21. Odds • What are the odds? • (1) Odds of A to B against an outcome means that the probability of that outcome is B / (A + B). • So, “odds of 5 to 1” is another way of saying “probability of 1/6.” • (2) Odds of C for a team winning means that the probability is C / (1 + C) • “Odds of 5 for a team winning” is another way of saying the “probability of the team winning is 5/6.” • Odds range from 0 to infinity.

  22. Odds • If the odds are 18 to 2 against a team winning, then the probability the team has of winning is estimated to be: • 2/16 = 0.125 • 2/18 = 0.111 • 2/20 = 0.100 • 16/18 = 0.889 • 18/20 = 0.900

  23. If the odds are 9 that a team wins, then the probability of the team winning is: • 9/10 = 0.900 • 9/18 = 0.5 • 1/9 = 0.111 • 8/9 = 0.889

  24. Chapter 18 – Probability Models probability model – describes all possible outcomes and says how to assign probabilities to any collection of outcomes sample space – collection of all unique outcomes of a random circumstance event – a collection of outcomes

  25. Coin Example Suppose you are asked to roll a die with 6 faces. What is the sample space? • Possible events are • Roll is an even number • Roll is an odd number • Roll is 5 or 6

  26. What about a roulette Sample space ? Example of events ?

  27. Probability Rules • Any probability is a number between 0 and 1. • So if we observe an event A then we know

  28. Probability Rules • 2. All possible outcomes together must have probability 1. • An outcome must occur on every trial. • The sum of the probabilities for all possible outcomes must be exactly 1.

  29. Marital Status of a Random Sample of Women • Consider the following assignment of probabilities • Marital Status of a Random Sample of Women Ages 25 to 29 Marital Status Probability Never married 0.386 Married 0.555 Widowed 0.004 Divorced 0.055

  30. Marital Status of a Random Sample of Women • Each of the probabilities is a number between 0 and 1. The probabilities total to 1. 0.386 + 0.555 + 0.004 + 0.055 = 1 Any assignment of probabilities to all individual outcomes that satisfies Rules 1 and 2 is legitimate.

  31. What does the probability of D need to be to make this a probability model? • P(A)=0.3 P(B)=0.2 P(C)=0.1 P(D)=? • 0.0 • 0.1 • 0.2 • 0.3 • 0.4

  32. Which of the following is not a possible probability model: • P(A)=0.3 P(B)=0.4 P(C)=0.3 • P(A)=0.3 P(B)=0.7 • P(A)=1.0 • P(A)=0.3 P(B)=0.6 P(C)=0.2

  33. Incoherent If a set of probabilities don’t satisfy rules 1 and 2 we say they are incoherent. This often occurs with someone’s personal probabilities in complicated situations

  34. Probability Rules • The probability that an event does not occur is 1 minus the probability that the event does occur. • This is known as the complement rule. • Suppose that P(A) = .70 • Using this rule we can determine P(not A) • P(not A) = 1- P(A) = 1-.70 = .30 The event “not A” is known as the complement of A which can be written as

  35. Suppose the probability of a horse winning a race is 0.85. What is the probability of the horse not winning? • A. 0.85 • B. 0.15 • C 0.7 • D 0.2

  36. Probability Rules • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. • If this is true then the events are said to be disjoint. Suppose events A and B are disjoint and you know that P(A) = .40 and P(B) = .35. What is the P(A or B)?

  37. Venn Diagrams

  38. If P(A)=0.5 and P(B)=0.4 and A and B are disjoint, then what is P(A or B)? • 0.1 • 0.2 • 0.4 • 0.5 • 0.9

  39. The probability a student is in honors math is 0.25, the probability a student is in honors science is 0.3, and the probability a student is in both is 0.2. • What is the probability a student is in at least one honors class?

  40. The probability it will rain Wednesday AM is 30%. The probability it will rain Wednesday PM is 30%. The probability it will rain both Wednesday AM and Wednesday PM is 10%. What is the probability it will rain on Wednesday? A) 20% D) 50% B) 30% E) 60% C) 40%

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