Transcript probability

Chapter 10
Introducing Probability
BPS - 5th Ed.
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Idea of Probability
 Probability
is the science of chance
behavior
 Chance behavior is unpredictable in the
short run but has a regular and
predictable pattern in the long run
– this is why we can use probability to gain
useful results from random samples and
randomized comparative experiments
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Randomness and Probability
 Random:
individual outcomes are
uncertain but there is a regular
distribution of outcomes in a large
number of repetitions
 Relative frequency (proportion of
occurrences) of an outcome settles down
to one value over the long run. That one
value is then defined to be the
probability of that outcome.
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Relative-Frequency Probabilities
 Can
be determined (or checked) by
observing a long series of independent
trials (empirical data)
– experience with many samples
– simulation (computers, random number
tables)
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Relative-Frequency Probabilities
Coin flipping:
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Probability Models
 The
sample space S of a random
phenomenon is the set of all possible
outcomes.
 An event is an outcome or a set of
outcomes (subset of the sample space).
 A probability model is a mathematical
description of long-run regularity
consisting of a sample space S and a way
of assigning probabilities to events.
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Probability Model for Two Dice
Random phenomenon: roll pair of fair dice.
Sample space:
Probabilities: each individual outcome has
probability 1/36 (.0278) of occurring.
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Probability Rule 1
Any probability is a number between
0 and 1.

A probability can be interpreted as the
proportion of times that a certain event can
be expected to occur.

If the probability of an event is more than 1,
then it will occur more than 100% of the time
(Impossible!).
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Probability Rule 2
All possible outcomes together must
have probability 1.

Because some outcome must occur on every
trial, the sum of the probabilities for all
possible outcomes must be exactly one.

If the sum of all of the probabilities is less
than one or greater than one, then the
resulting probability model will be incoherent.
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Probability Rule 3
If two events have no outcomes in common,
they are said to be disjoint. The probability
that one or the other of two disjoint events
occurs is the sum of their individual
probabilities.
 Age of woman at first child birth
– under 20: 25%
24 or younger: 58%
– 20-24: 33%
– 25+: ? Rule 3 (or 2): 42%
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Probability Rule 4
The probability that an event does
not occur is 1 minus the probability
that the event does occur.

As a jury member, you assess the probability
that the defendant is guilty to be 0.80. Thus
you must also believe the probability the
defendant is not guilty is 0.20 in order to be
coherent (consistent with yourself).
 If
the probability that a flight will be on time is
.70, then the probability it will be late is .30.
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Probability Rules:
Mathematical Notation
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Probability Rules:
Mathematical Notation
Random phenomenon: roll pair of fair dice and
count the number of pips on the up-faces.
Find the probability of rolling a 5.
P(roll a 5) = P(
= 1/36
)+P(
)+P(
)+P(
+
+ 1/36
+ 1/36
1/36
= 4/36
= 0.111
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 Exercise
10.10: Choose a Canadian at
random and ask “What is your mother
tongue?” Here is the distribution of
responses:
 English – Prob. 0.63, French – 0.22,
Asian/Pacific – 0.06, Other - ?
 a) What prob. should replace ?
 B) What is the prob. that a Canadian’s
mother tongue is not English
 C) What is the prob. that a Canadian’s
mother tongue is a language other than
English
and
French?
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 10.31:Describe
the sample space S for each
of the following random phenomenon.
 A) A basketball player shoots four free
throws. You record the sequene of hits and
misses.
 B) A basketball player shoots four free
throws. You record the number of baskets
she makes.
 C) Can you assign probabilities to all the
events/ outcomes in the previous problems?
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 Exercise
10.36: Here are the probabilities of
the most popular colors for vehicles in North
America
 White: 0.19, Silver: 0.18, Black: 0.16,
Red:0.13, Gray:0.12 , Blue:0.12
 A) what is the probability that a vehicle you
choose has any color other than the six
listed?
 B) what is the probability that a randomly
chosen vehicle is neither silver nor white?
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Discrete Probabilities
 Finite
(countable) number of outcomes
– assign a probability to each individual
outcome, where the probabilities are
numbers between 0 and 1 and sum to 1
– the probability of any event is the sum of
the probabilities of the outcomes making
up the event
– see previous slide for an example
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Continuous Probabilities
 Intervals
of outcomes
– cannot assign a probability to each individual
outcome (because there are an infinite number
of outcomes)
– probabilities are assigned to intervals of
outcomes by using areas under density curves
– a density curve has area exactly 1 underneath
it, corresponding to total probability 1
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Assigning Probabilities:
Random Numbers Example
Random number generators give output (digits)
spread uniformly across the interval from 0 to 1.
Find the probability of
getting a random number
that is less than or equal
to 0.5 OR greater than 0.8.
P(X ≤ 0.5 or X > 0.8)
= P(X ≤ 0.5) + P(X > 0.8)
= 0.5 + 0.2
= 0.7
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Normal Probability Models
 Often
the density curve used to assign
probabilities to intervals of outcomes is the
Normal curve
– Normal distributions are probability models:
probabilities can be assigned to intervals of
outcomes using the Standard Normal
probabilities in Table A of the text (pp. 690-691)
– the technique for finding such probabilities is
found in Chapter 3
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Normal Probability Models
Example: convert
observed values of
the endpoints of the
interval of interest to
standardized scores
(z scores), then find
probabilities from
Table A.
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Random Variables
A
random variable is a variable whose value
is a numerical outcome of a random
phenomenon
– often denoted with capital alphabetic symbols
(X, Y, etc.)
– a normal random variable may be denoted as
X ~ N(µ, )
 The
probability distribution of a random
variable X tells us what values X can take and
how to assign probabilities to those values
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Random Variables
 Random
variables that have a finite
(countable) list of possible outcomes,
with probabilities assigned to each of
these outcomes, are called discrete
 Random
variables that can take on any
value in an interval, with probabilities
given as areas under a density curve, are
called continuous
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Random Variables
 Discrete
random variables
– number of pets owned (0, 1, 2, … )
– numerical day of the month (1, 2, …, 31)
– how many days of class missed
 Continuous
random variables
– weight
– temperature
– time it takes to travel to work
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exercise 10.49 and 10.50: A random
number Y is chosen between 0 and 2.
 A) Is the random variable Y discrete or
continuous? If it is continuous, can you
describe the density curve?
 B) Find P(Y<=1)
 C) Find P(0.5<Y<1.3)
 D) Find P(Y>=0.8)

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 Exercise
10.18: A study of 12000 able
bodied male students in U of Illinois found
that their times for the mile run were
approximately Normal with mean 7.11
minutes and standard deviation 0.74.
Choose a student at random from this
group and call his time for the mile Y.
 A) Find P(Y>=8)
 B) what is the event “the student could run
a mile in less than 6 minutes” in terms of
Y? Find the prob. Of this event.
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Personal Probabilities
 The
degree to which a given individual
believes the event in question will happen
 Personal belief or judgment
 Used to assign probabilities when it is not
feasible to observe outcomes from a long
series of trials
– assigned probabilities must follow established
rules of probabilities (between 0 and 1, etc.)
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Personal Probabilities
 Examples:
– probability that an experimental (never
performed) surgery will be successful
– probability that the defendant is guilty in a
court case
– probability that you will receive a ‘B’ in this
course
– probability that your favorite baseball team
will win the World Series in 2020
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