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Statistics 303
Chapter 4 and 1.3
Probability
Probability
• The probability of an outcome is the proportion
of times the outcome would occur if we repeated
the procedure many times.
• Examples
–
–
–
–
Coin: What is the probability of obtaining heads when flipping a coin?
A single die: What is the probability I will roll a four?
Two dice: What is the probability I will roll a four?
A jar of 30 red and 40 green jelly beans: What is the probability I will
randomly select a red jelly bean?
– Computer: In the past 20 times I used my computer, it crashed 4 times and
didn’t crash 16 times. What is the probability my computer will crash
next time I use it?
Probability
• Independence: Two events are independent
if the outcome of one does not affect or give
an indication of the outcome of the other.
Dependent
Events
Independent
Flipping a coin twice
Temperature on
consecutive days
3 jelly beans: red,
green, orange. Eat
one. Eat another.
Probability
• Independence: Two events are independent
if the outcome of one does not affect or give
an indication of the outcome of the other.
Events
Randomly polling
two individuals
Comparing fertilizer
yield for two adjacent
field plots
Rolling two dice
Independent
Dependent
Probability
• Definition: A sample space is a set of all
the possible outcomes of a process.
– Example: Coin
• What is the sample space for flipping a coin 3
times?
Probability
• Definition: An event is an outcome or set of
outcomes of a process.
– Example: Coin
• What is one of the possible events for flipping a coin
3 times?
Probability Rules
• Rule 1: The probability of any event is between 0
and 1 inclusive.
– Pr(HTH) = 1/8 which is between 0 and 1.
• Rule 2: The probability of the whole sample space
is 1.
– Pr(rolling a 1 or 2 or 3 or 4 or 5 or 6) = 1
• Rule 3: The probability of an event not occurring
is 1 minus the probability of the event. This is
known as the complement rule.
– Pr(not rolling a 5) = 1 – 1/6 = 5/6
Probability Rules
• Rule 4: If two events A and B have no outcomes in
common (they are disjoint), then Pr(A or B) = Pr(A) +
Pr(B)
– Pr(rolling a 1 or a 6) = Pr(rolling a 1) + Pr(rolling a 6) = 1/6
+ 1/6 = 2/6 or 1/3
• Rule 5: If two events A and B are independent, then
Pr(A and B) = Pr(A)Pr(B)
– Pr(rolling a 1 and then a 6) = Pr(rolling a 1) * Pr(rolling a 6)
= (1/6)(1/6) = 1/36
Rules of Probability
• Rule 1: 0 ≤ P(A) ≤ 1
• Rule 2: P(S) = 1
• Rule 3: Complement Rule: For any event, A,
P(Ac) = 1 – P(A)
• Rule 4: Addition Rule: If A and B are disjoint
events, then
P(A or B) = P(A) + P(B)
• Rule 5: Multiplication Rule: If A and B are
independent events, then
P(A and B) = P(A)P(B)
Random Variables
• A random variable is a variable whose value
is a numerical outcome of a random
phenomenon.
• A discrete random variable, X, has a finite
number of possible values. The probability
distribution of X lists the values and their
probabilities.
Discrete Probability Example
X
P(X)
1
.1
2
.1
3
.2
4
.3
5
.3
• The table above represents the distribution of a
discrete random variable, X. How likely are you
to get an X more than 2?
• How likely are you to get TWO 3’s if you take a
random sample of 2 from this population?
Discrete Probability Example
X
P(X)
1
0.1
2
0.1
3
0.2
4
0.3
5
0.3
• To find the mean of a discrete distribution, multiply each
possible value by its probability, then add all the products.
• To find the variance of a discrete distribution, subtract the mean
from each of the X’s, square it, multiply it by the corresponding
probability, then add up all the products.
Random Variables
• A continuous random variable, X, takes all values
in an interval of numbers. The probability
distribution is described by a density curve. The
probability of any event is the area under the
density curve and above the values of X that make
up the event.
• Examples are uniform, normal, left-skewed
distributions, and right-skewed distributions.
Uniform Distribution Example
• This is a Uniform Distribution from 0 to 1. Since
the area under the curve is 1, the height is also 1.
To find the probability for a given interval, you
find the areas under the curve.
The Normal Distribution
The mean for a normal distribution is called
μ (pronounced ‘myu’), the Greek letter for
m (for mean). The standard deviation for a
normal distribution is called s (pronounced
‘sigma’), the Greek letter for s (for std. dev.)
s
m
The Normal Distribution
We often write that a variable (call it X) has
normal distribution with mean m and variance
s2 in the following way:

X ~ N m ,s
2

Note that the std.
dev. is still s.
s
m
The Normal Distribution
The normal distribution we have already
seen is the standard normal distribution,
which has mean = 0 and standard deviation
= 1 (the variance = 12 = 1). This is also
called the Z-distribution.
1
0
The Normal Distribution
The distribution of any variable which is
normally distributed can be converted to a
standard normal (Z) distribution in the
following way:
Z 
X m
s
This is known as a Z-score.
The Normal Distribution
The distribution of any variable which is
normally distributed can be converted to a
standard normal (Z) distribution in the
following way:
X m

X ~ N m ,s 2
Z 

~ N 0,1
s
s
m
1
X
0
The Normal Distribution
• For the Standard Normal Distribution (or
Z-Distribution) we can find probabilities
associated with different values of Z using
Z-tables.
Z
The Normal Distribution
• First we look at some general characteristics
of the Z-distribution.
–
–
–
–
The area under the entire curve is 1.
The area under the curve to the left of 0 is 0.5.
We say, “The probability that Z is to the left of 0 is 0.5.”
This can be written as Prob ( Z < 0) = 0.5.
1
0.5
Z
0
The Normal Distribution
• We can find the probability that Z is to the
left of any number using the Z-table.
– Z-tables can be found at http://stat.tamu.edu/stat30x/zttables.html
– Z-tables can also be found on the inside front cover of the book
– Notice first if we go in the table to the value z = 0.00 we see the
probability is 0.5.
0.5
Z
0
The Normal Distribution
• We can find the probability that Z is to the
left of any number using the Z-table.
See explanation at http://stat.tamu.edu/stat30x/notes.html called How To
Use Z Tables
– Let’s look at some examples:
Pr ( Z < 0.50) = ?
Pr ( Z < 1.25) = ?
Answer
Answer
Z
Z
1.25
0.50
The Normal Distribution
• More examples of probabilities to the left or
less than a number
Pr ( Z < -3.75) = ?
Pr ( Z < -2.01) = ?
Answer
Answer
Z
-2.01
Z
-3.75
The Normal Distribution
• The Z-table only gives probabilities to the left of
the value. If we want to get probabilities to the
right we use 1 – Pr (Z < z).
Pr ( Z > 0.50) = ?
Pr ( Z > 1.25) = ?
Answer:
Answer:
Z
Z
1.25
0.50
The Normal Distribution
• More examples of finding probabilities to the right
of a number using 1 – Pr (Z < z).
Pr ( Z > -3.75) = ?
Pr ( Z > -2.01) = ?
Answer: >
Answer:
Z
-2.01
Z
-3.75
The Normal Distribution
• To find probabilities between two numbers,
find the less than (of to the left) probability
for each number and then subtract.
Pr (-2.01< Z < 2.01) = ?
ANSWER:
The Normal Distribution
• Now suppose we know X ~ N (m, s2) and we want
to know the probability that X is less than some
value. We must first convert the X to a Z and then
use the probabilities from the Z-table. Recall that
if X ~ N (m, s2) , then
Z 
X m
s
~ N 0,1
So Pr (X < x) = Pr (Z < (x – m)/s
The Normal Distribution
• Here’s an example. Suppose X ~ N ( 3, 22). Find
the probability that X is less than 4.
Pr ( X < 4 ) = ?
Z
Pr ( X < 4 ) = Pr (Z < 0.5) =
X m
s

43
 1 / 2  0.5
2
Z
X
3
4
0
0.50
The Normal Distribution
• We will look at some more difficult
examples by hand:
– Suppose X ~ N (2, 32),
•
•
•
•
•
•
Given a value z, find the corresponding x that it came from.
How many standard deviations is x from m?
Find Pr (X > 5).
Find Pr (X < –4 or X > 8).
Find Pr ( –4 < X < 8 ).
Find the x* such that Pr ( X < x* ) = 0.8485, where 0.8485 is
some probability.
Normal Distribution Example
• Suppose the sample proportion of 100
students who think that there is insufficient
parking is normally distributed with a mean
of 0.8 and a standard deviation of 0.04 ie. p
~ N(0.8, 0.042).
How often would we get a sample
proportion of 0.75 or less?
Law of Large Numbers
• Draw independent observations at random from
any population with finite mean, μ. Decide how
accurately you would like to estimate μ. As the
number of observations drawn increases, the
sample mean ( x ) of the observed values eventually
approaches the mean μ of the population as closely
as you specified and then stays that close.
Rules for means
Rule 1: If X is a random variable and a and b are fixed
numbers, then
ma bX  a  bm X
Rule 2: If X and Y are random variables, then
m X Y  m X  mY
Rules for Variances
Rule 1: If X is a random variable and a and b are fixed
numbers, then
s 2 a bX  b 2s 2 X
Rule 2: If X and Y are independent random variables, then
s 2 X Y  s 2 X  s Y2
Note: variances, NOT std. devs. always add. If X and Y
are not independent, then there is a correlation factor added
or subtracted from their sum.
Variance when X and Y are
Correlated
Rule 2b: If X and Y are NOT independent
random variables, then
s
2
s
2
X Y
s
2
X Y
s
2
X
 s  2 s xs y
X
 s  2 s xs y
2
Y
2
Y
where  is the correlation factor between X and Y.
General Addition Rules
• The union of any collection of events is the event that at
least one of the collection occurs.
• Addition Rule for Disjoint Events – If events A, B, and C
are disjoint in the sense that no two have any outcomes in
common, then
P(one or more of A, B, C) = P(A)+P(B)+P(C)
• General Addition Rule for the Unions of Two Events – For
any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)