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