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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)
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? …
Example, p.25 (b)
Quantity Demanded (Gallons)
Number of DaysFrequency as Probability
0 40
1 80
2 50
3 20
4 10
Total
‘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
• (your education; your starting salary)
• (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, 1 for head)
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):
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.
)(...)()(
)()(
2211
1
nn
n
iii
XPXXPXXPX
XPXXE
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
n
iii XPXXE
1
)()(
Xi P(Xi)
i X’s possible value Probability
1 5 0.1
2 4 0.2
3 3 0.3
4 2 0.3
5 1 0.1
X=a student’s quiz score
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 = = 2 variance
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
• 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.
n
iii XPXEX
1
22 )()]([
Calculating 2 in Example p.34
Xi P(Xi)
i X’s possible value Probability
1 5 0.1
2 4 0.2
3 3 0.3
4 2 0.3
5 1 0.1
X=a student’s quiz score
n
iii XPXEX
poncalculatedasXE
1
22 )()]([
.35.9.2)(
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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