Probability Concepts and Applications

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Chapter 2

• Probability Concepts and Applications

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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)

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Basic Laws of Probabilities

• 0 lt P(event) lt 1
• Sum of the probabilities of all possible outcomes
  of an activity (a trial) equals to 1.

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Subjective Probability

• Subjective Probability is coming from persons
  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.

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Objective Probability

• Objective Probability is the frequency that is
  derived from the past records
• How to calculate frequency?
• Example page 25 and page 34

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Example, p.25 (a)Calculate probabilities of
daily demand from data in the past
Demanded daily (Gallons) Number of Days
0 40
1 80
2 50
3 20
4 10

What is the probability that daily demand is 4 gallons? 3 gallons? 2 gallons? What is the probability that daily demand is 4 gallons? 3 gallons? 2 gallons?
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Example, p.25 (b)
Demanded Daily (Gallons) Number of Days Frequency as Probability
0 40
1 80
2 50
3 20
4 10


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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.

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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.

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Intersection of Events

• Intersection of two events A and B refers to (A
  and B), which is also put as AnB or simply AB.
• For example, drawing one from 52 playing cards.
  If A a 7 is drawn, B a heart is drawn, then
  AnB means the card drawn is 7 and a heart.

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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(AB).
• For example,
• P(7 Heart).
• P(battery is bad engine wont start)

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Formulas for U and n

• P(AUB) P(A) P(B) ? P(AnB)
• P(AnB) P(A)P(BA)
• by algebraic rule we have
• P(BA) P(AnB) / P(A)

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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(AnB) P(AB) 1/52.
• P (AUB) 4/52 13/52 ? 1/52 16/52
• P(AB) P(AB) / P(B) 1/52 / 13/52
• 1/13.

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Mutually Exclusive Events

• Two events are mutually exclusive if they cannot
  both occur.
• A few events are mutually exclusive if only one
  can occur.
• If A and B are mutually exclusive, then
• P(AnB) 0.

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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)

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Probabilities for Mutually Exclusive Events

• If events A and B are mutually exclusive, then
• P(AUB) P(A) P(B)

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Independent Events

• Two events are independent if the occurrence of
  one event has no effect on the probability of
  occurrence of the other i.e., they are not
  related.
• If A and B are independent, then
• P(AB) P(A), and P(BA) P(B).

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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)

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Examples for Non-Independent Events

• (education starting salary)
• (it rains today there are thunders today)
• (heart disease diabetes)
• (losing control of a car the driver is drunk).

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Formulas for P(AnB) if A and B Are Independent

• If A and B are independent, then their joint
  probability formula is reduced to
• P(AnB) P(A) P(B)

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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(BG)
• P(BB)
• P(GG)
• P(GBB)

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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(BG)
• P(BB)
• P(GB)
• P(GB)

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Discerning between Mutually Exclusive and
Independent

• A and B are mutually exclusive if A and B cannot
  both occur. P(AnB)0.
• A and B are independent if As occurrence has no
  influence on the chance of Bs occurrence, and
  vice versa. P(AB)P(A) and P(BA)P(B).

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Discerning Conditional Probability and Joint
Probability

• Joint probability P(AB) or P(AnB) is the chance
  both A and B occur before either actually occurs.
• Conditional probability P(AB) is the chance of A
  after knowing that B has occurred.

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Random Variable

• A random variable is such a variable whose value
  is selected randomly from a set of possible
  values.

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Examples of Random Variables

• Z outcome of tossing a coin (0 for tail, 1 for
  head)
• Xnumber of refrigerators sold a day
• Xnumber of tokens out for a token you put into a
  slot machine
• YNet profit of a store in a month
• Table 2.5 and 2.6, p.33

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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.37.

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Expected Value of X

• The expected value of random variable X is the
  average of the possible values that X may take.

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Formula for Expected Value

• The expected value of X E(X)
• where Xithe i-th possible value of X,
• P(Xi)probability of Xi,
• nnumber of possible values.
• E(X) is the sum of Xs possible values weighted
  by their probabilities.

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Xi, P(Xi), and E(X) in Example p.34
Xa students quiz score
Xi P(Xi)
i Xs possible value Probability
1 5 0.1
2 4 0.2
3 3 0.3
4 2 0.3
5 1 0.1
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Other Examples

• Expected value of a game of tossing a coin.
• Expected value of playing with a slot machine
  (see the handout).

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Standard Deviation of X

• Standard deviation (SD), ?, of random variable X
  is the average of the distances between Xs
  possible values X1, X2, X3, and Xs expected
  value E(X).

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Variance of X

• To calculate standard deviation (SD), we need to
  first calculate variance.
• Variance is the average of the squared distances
  between Xs possible values and E(X).
• Variance ?2 (SD)2.
• SD ?

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Standard Deviation and Variance

• Both SD and variance are parameters showing the
  spread or dispersion of possible values of X.
• The larger the SD and variance, the more
  dispersed the distribution.

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Calculating Variance ?2

• where
• ntotal number of possible values,
• Xithe i-th possible value of X,
• P(Xi)probability of the i-th possible value
  of X,
• E(X)expected value of X.

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Calculating ?2 in Example p.34
Xa students quiz score
Xi P(Xi)
i Xs possible value Probability
1 5 0.1
2 4 0.2
3 3 0.3
4 2 0.3
5 1 0.1
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E(X), SD, and Variance

• E(X) is Expected Value of X, which is the average
  of Xs possible values.
• SD is the average of the distances between Xs
  possible values and E(X).
• Variance of X is the average of the squared
  distances between Xs possible values and E(X).

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What If SD0

• Variance and SD cannot be negative.
• The smallest value of SD or variance is 0.
• SD0 or variance0 means zero deviation of Xs
  possible values, which indicates that X takes
  some value always so X is no longer a random
  variable.

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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 ?.

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Standard Normal Distribution

• The standard normal distribution has the
  parameters ?0 and ?1.
• Symbol Z denotes the random variable with the
  standard normal distribution