STAT 110 - Section 5 Lecture 20

Professor Hao WangUniversity of South Carolina

Spring 2012

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%

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.

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.

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.

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

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.

Myths About Chance

1. 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.

Example

What looks random?

Toss a coin six times. Which of these outcomes is more probable?

HTHTTH TTTHHH

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.

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?

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.

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!

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.

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.

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?

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?

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

http://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.

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.

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.

Odds

If the odds are 18 to 2 against a team winning, then the probability the team has of winning is estimated to be:

A) 2/16 = 0.125

B) 2/18 = 0.111

C) 2/20 = 0.100

D) 16/18 = 0.889

E) 18/20 = 0.900

• If the odds are 9 that a team wins, then the probability of the team winning is:

A)9/10 = 0.900

B)9/18 = 0.5

C)1/9 = 0.111

D)8/9 = 0.889

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

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

What about a roulette

Sample space ?

Example of events ?

Probability Rules

1. Any probability is a number between 0 and 1.

So if we observe an event A then we know

1)(0 AP

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.

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

• 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.

Marital Status of a Random Sample of Women

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)=?

A) 0.0

B) 0.1

C) 0.2

D) 0.3

E) 0.4

Which of the following is not a possible probability model:

A) P(A)=0.3 P(B)=0.4 P(C)=0.3

B) P(A)=0.3 P(B)=0.7

C) P(A)=1.0

D) P(A)=0.3 P(B)=0.6 P(C)=0.2

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

Probability Rules

3. 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

cA

• 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

Probability Rules

4. 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)?

Venn Diagrams

If P(A)=0.5 and P(B)=0.4 and A and B are disjoint, then what is P(A or B)?

A) 0.1

B) 0.2

C) 0.4

D) 0.5

E) 0.9

• 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?

• 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%