Download STAT 155 Introductory Statistics Lecture 13: Birthday Problem

The UNIVERSITY of NORTH CAROLINA
at CHAPEL HILL
STAT 155 Introductory Statistics
Lecture 13: Birthday Problem, Prisoner
Dilemma, Random Variable
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Review
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Birthday Problem
• In a classroom of 45 people, what is the
probability that at least two people have the
same birthday?
• Event A: at least two people have the same
birthday out of the 45 people.
• AC: every person has a different birthday
out of the 45 people.
• P(A) = 1 - P(AC) = … (see the board)
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3 Prisoners’ Dilemma
• Three prisoners A, B and C are on a death row. The
governor decides to pardon one of the three and
chooses at random the prisoner to pardon.
• He informs the warden of his choice but requests
that the name be kept secret for a few days.
• The next day, A tries to get the warden to tell him
who had been pardoned.
• The warden refuses. A then asks which of B or C will
be executed. The warden thinks for a while, then
tells A that B is to be executed.
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3 Prisoners’ Dilemma
A's reaction: Given that B will be executed,
then either A or C will be pardoned, my
chance of being pardoned has increased
from 1/3 to 1/2.
Q: Is A correct ? Did the warden disclose any
information to A ? Can A increase his
chance of survival by swapping with C ?
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Free throws
• A TarHeel basketball player is a 80% free
throw shooter.
• Suppose he will shoot 20 free throws
during each practice.
• Which is more likely: to make 5 out of 20,
or 18 out of 20 ?
• How many free throws he makes on
average during practice?
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Random Variables
• Experiment:
– A TarHeel basketball player shoots 20 free
throws during his practice.
– X: number of hits
• A random variable is a variable whose value
is a numerical outcome of a random
experiment.
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Two types of random variables
• A discrete random variable has a finite
number of possible values.
– X: number of hits when trying 20 free throws.
– Possible values for X: 0,1, …, 20
• A continuous random variable takes
values in an interval.
– X: the time it takes for a bulb to burn out.
– Possible values are not countable.
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Discrete Random Variable
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Flip a coin 4 times
• Find the probability distribution of the random variable
describing the number of heads that turn up when a
fair coin is flipped 4 times.
• Solution
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Probability Histogram
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Questions
• What is the connection with histograms we
talked about in Chapter 1?
• Are the two problems similar (toss a coin 4
times and shoot 20 free throws) ?
Yes or no … the ``free throw’’ problem is
equivalent to tossing a biased coin 20
times, each with P(H) = 0.8.
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Continuous Random Variable (spinner)
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Continuous Random Variable
• A continuous random variable X takes all
possible values in an interval.
– Not countable
• The probability distribution of a continuous r.v. X
is described by a density curve.
– What is a density curve?
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Ex: Spinner (continued)
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•
•
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P( point to 1/4) = 0 (why ?)
P( greater than 5/8) = 1 – 5/8 = 3/8
P( between 2/9 and 7/8) = 7/8 – 2/9 = …
P( falling in (x, x+1/4)) = 1/4 for any x
greater than 0 and less than 3/4.
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Continuous Distribution
• The probability of any event is the area under the density
curve and above the values of X that make up the event.
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Continuous Distribution
• The probability model for a continuous
random variable assigns probabilities to
intervals of outcomes rather than to
individual outcomes.
• In fact, all continuous probability
distributions assign probability 0 to every
individual outcome.
– The spinner
• Normal distributions are continuous
probability distributions.
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Women Height
• The height of American women aged 18 – 24 is
approximately normally distributed with mean 64.3
inches and s.d. 2.4 inches. Two women in the age
group are randomly selected.
• What is the probability that both of them are taller than
66 inches?
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