# Example

```Chapter 2
Probability Concepts and
Applications
Probability
• A probability is a numerical description of the
chance that an event will occur.
• Examples:
P(it rains tomorrow)
P(flooding in St. Louis in September)
P(winning a game at a slot machine)
P(50 or more customers coming to the store in the next
hour)
P(A checkout process at a store is finished within 2
minutes)
Basic Laws of Probabilities
• 0 <= P(event) <= 1
• Sum of the probabilities of all possible
outcomes of an activity (a trial) equals to 1.
Subjective Probability
• Subjective Probability is coming from
person’s judgment or experience.
• Example:
– Probability of landing on “head” when tossing a
coin.
– Probability of winning a lottery.
– Chance that the stock market goes down in
coming year.
Objective Probability
• Objective Probability is the frequency
that is derived from the past records
• How to calculate frequency?
– Example: page 23 and page 34
Example, p.25 (a)
Calculate probabilities of daily demand from data in the past
Quantity Demanded
(Gallons)
0
1
2
3
4
Number of Days
40
80
50
20
10
What is the probability that daily demand is 4 gallons? 3 gallons? 2 gallons? …
Example, p.25 (b)
Quantity
Demanded
(Gallons)
0
1
2
3
4
Total
Number of Days
40
80
50
20
10
Frequency as
Probability
‘Possible Outcomes’ vs.
‘Occurrences’
• In the given data, differentiate the column
for ‘possible outcomes’ of an event from the
column for ‘occurrences’ (how many times
an outcome occurred).
• Probabilities are about possible ‘outcomes’,
whose calculations are based on the column
of ‘occurrences’.
Union of Events
• Union of two events A and B refers to
(A or B), which is also put as AUB.
• For example, drawing one from 52 playing
cards. If A= a “7” is drawn, B= a “heart” is
drawn, then AUB means “the card drawn is
either a ‘7’ or a ‘heart’”.
Intersection of Events
• Intersection of two events A and B
refers to (A and B), which is also put
as A∩B or simply AB.
• For example, drawing one from 52 playing
cards. If A= a “7” is drawn, B= a “heart” is
drawn, then A∩B means “the card drawn is
‘7’ and a ‘heart’”.
Conditional Probability
• A conditional probability is the
probability of an event A given that
another event B has already happened.
• It is put as P(A|B).
• For example,
– P(a man has got cancer | his PSA test value is
1.5),
– P(battery is dead | engine won’t start)
Formulas for U and ∩
• P(AUB) = P(A) + P(B)  P(A∩B)
• P(A∩B) = P(A) * P(B|A)
by algebraic rule we have
P(B|A) = [P(A∩B)] / P(A)
Example (p.27-28)
• Randomly draw one from 52 playing cards.
Let A= a “7” is drawn, B= a “heart” is
drawn:
• P(A) = 4/52, P(B) = 13/52,
• P(A∩B) = P(AB) = 1/52.
• P (AUB) = 4/52 + 13/52  1/52 = 16/52
• P(A|B) = [P(AB)] / P(B) = [1/52] / [12/52]
= 1/13.
Mutually Exclusive Events
• Events are mutually exclusive if only one of
the events can occur on any trial.
• If A and B are mutually exclusive, then
P(A∩B) = 0.
Examples
• Mutually exclusive:
– (it rains at AC; it does not rain at AC)
– Result of a game: (win, tie, lose)
– Outcome of rolling a dice: (1, 2, 3, 4, 5, 6)
• NOT mutually exclusive:
– (a randomly drawn card is a ‘7’; a randomly
drawn card is a ‘heart’.)
– (one involves in an accident; one is hurt in an
accident)
Probabilities for Mutually
Exclusive Events
• If events A and B are mutually
exclusive, then:
P(AUB) = P(A) + P(B)
Independent Events
• Two events are independent if the
occurrence of one event has no effect
on the probability of occurrence of the
other.
• If A and B are independent, then
P(A|B) = P(A), and P(B|A) = P(B).
Examples of Independent Events
• (results of tossing a coin twice)
• (lose \$1 in a run on a slot machine, lose
another \$1 in the next run on the slot
machine)
• (it rains at AC; it does not rain at LA)
Examples for Non-Independent
Events
•
•
•
•
(it rains today; there are thunders today)
(heart disease; diabetes);
(losing control of a car; the driver is drunk).
Formulas for P(A∩B) if A and
B Are Independent
• If A and B are independent, then their
joint probability formula is reduced to:
P(A∩B) = P(A) * P(B)
Example
• Drawing balls one at a time with
replacement from a bucket with 2 blacks
(B) and 3 greens (G).
–
–
–
–
–
–
Is each drawing independent of the others?
P(B) =
P(B|G) =
P(B|B) =
P(GG) =
P(GBB) =
Example
• Drawing balls one at a time without
replacement from a bucket with 2 blacks
(B) and 3 greens (G).
–
–
–
–
–
–
Is each drawing independent of the others?
P(B) =
P(B|G) =
P(B|B) =
P(GB) =
P(G|B) =
Discerning between Mutually
Exclusive and Independent
• A and B are mutually exclusive if A
and B cannot both occur. P(A∩B)=0.
• A and B are independent if A’s
occurrence has no influence on the
chance of B’s occurrence, and vice
versa. P(A|B)=P(A) and P(B|A)=P(B).
Discerning Conditional
Probability and Joint Probability
• Joint probability P(AB) or P(A∩B) is
the chance both A and B occurs before
either actually occurs.
• Conditional probability P(A|B) is the
chance of A after knowing that B has
occurred.
Random Variable
• A random variable is such a variable
whose value is selected randomly from
a set of possible values.
Examples of Random Variables
 Z = outcome of tossing a coin (0 for tail,
 X=number of refrigerators sold a day
 X=number of tokens out for a token you
put into a slot machine
 Y=Net profit of a store in a month
 Table 2.5 and 2.6, p.33
Probability Distribution
• The probability distribution of a
random variable shows the probability
of each possible value to be taken by
the variable.
• Example: P.34, P.35, P.38.
Expected Value of a Random
Variable X
• The expected value of X = E(X):
n
E ( X )   X i P( X i )
i 1
 X 1 P( X 1 )  X 2 P( X 2 )  ...  X n P( X n )
where Xi=the i-th possible value of X,
P(Xi)=probability of Xi,
n=number of possible values.
• E(X) is the sum of X’s possible values weighted
by their probabilities.
Interpretation of Expected
Value
• The expected value is the average
value (mean) of a random variable.
Xi, P(Xi), and E(X) in Example p.34
X=a student’s quiz score
i
1
2
3
4
5
Xi
X’s possible value
5
4
3
2
1
n
E ( X )   X i P( X i ) 
i 1
P(Xi)
Probability
0.1
0.2
0.3
0.3
0.1
Other Examples
• Expected value of a game of tossing a coin.
• Expected value of playing with a slot
machine (see the handout).
Standard Deviation of X
• Standard deviation (SD), , of random
variable X is the average distance of
X’s possible values X1, X2, X3, …
from X’s expected value E(X).
Variance of X
• To calculate standard deviation (SD),
we need to first calculate “variance”.
• Variance 2 = (SD)2.
• SD =  = variance   2
Standard Deviation and
Variance
• Both standard deviation and variance
are parameters showing the spread or
dispersion of the distribution of a
random variable.
• The larger the SD and variance, the
more dispersed the distribution.
Calculating Variance 2
n
  [ X i  E ( X )] P( X i )
2
2
i 1
• where
• n=total number of possible values,
• Xi=the i-th possible value of X,
• P(Xi)=probability of the i-th possible value of X,
• E(X)=expected value of X.
Calculating 2 in Example p.34
X=a student’s quiz score
Xi
X’s possible value
5
4
3
2
1
i
1
2
3
4
5
E ( X )  2.9
n
as calculatedon p.35.
 2   [ X i  E ( X )]2 P( X i ) 
i 1
P(Xi)
Probability
0.1
0.2
0.3
0.3
0.1
Normal Distribution
• The normal distribution is the most popular
and useful distribution.
• A normal distribution has two key
parameters, mean  and standard deviation
.
• A normal distribution has a bell-shaped
curve that is symmetrical about the mean .
Standard Normal Distribution
• The standard normal distribution has the
parameters =0 and =1.
• Symbol Z denotes the random variable with
the standard normal distribution
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