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STAT 110 - Section 5
Lecture 20
Professor Hao Wang
University 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.h
tml
http://reason.com/archives/2006/08/11/dont-be-terrorized
http://www.independent.co.uk/news/world/americas/us-living-infear-over-summer-of-child-abductions-649895.html
http://finance.yahoo.com/news/never-safer-flydeaths-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
0  P( A)  1
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
Never married
Married
Widowed
Divorced
Probability
0.386
0.555
0.004
0.055
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.
What does the probability of D need to be to make
this a probability model?
P(A)=0.3
A) 0.0
B) 0.1
C) 0.2
D) 0.3
E) 0.4
P(B)=0.2
P(C)=0.1
P(D)=?
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
c
A
• 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%