C HAPTER 5:P ROBABILITY
1.D EFINITIONS
Experiment :a process that produces outcomes.
–More than one possible outcome
–Only one outcome per trial
Trial : one repetition of the process
Example 1a.
Interviewing 20 randomly lected consumers and asking them which brand of appliance they prefer.
Example 2a.
Randomly lect, without replacement, two families from the residents of Tiny Town.
Tiny town population
Family Children in houhold Number of automobiles
A Yes 3
B Yes 2车尘马足
C No 1
D Yes 2
Example 3a.
Sample five bottles coming off a production line.
Elementary Event : cannot be decompod or broken down into other events
Event : an outcome of an experiment
–may be an elementary event, or
–may be an aggregate of elementary events
–usually reprented by an upperca letter, i.e. A, E1
Example 1b.
小红帽和大灰狼
Elementary event : people who prefer Sony.
Joint event : people who prefer Sony and Toshiba.
Example 2b.
-Elementary event : the sample includes families A and C
-Joint event : each family in the sample has children in the houhold
Example 3b.
Event : get one defective and four good bottles
Sample space : a complete roster or listing of all elementary events for an experiment. Examples.
草莓饭团- All six faces of a die.
-
All 52 card of a deck.
Probability : it’s the numerical measure of the likelihood that an event will occur.
Example 4 :
Event A : get four good bottles in a production line
The probability of that event will occur is denoted P(A).
Probability of the complement
2. A SSIGNING PROBABILITY TO EVENTS
We are going to introduce the notion of a random experiment and describe the outcomes or events that may result from such an experiment.
When attempting to solve any problem involving probabilities, you should begin by defining the random experiment and the sample space.
Probabilities can be assigned to the simple events using :
- the classical approach (rules and laws) :
With n e = the number of outcomes of an experiment
and N = Total possible number of outcomes of an experiment.
- the relative frequency approach (cumulated historical data) :
P(E) =occur event to for the ies opportunit of number Total occured event an times of Number = T
X - the subjective approach (personal intuition or reasoning).
Whatever method is ud to assign probabilities to the simple events that form a sample space, two basic requirements must be satisfied:
1. Each simple event probability must lie between 0 and 1, inclusive.
2. The probabilities assigned to the simple events in a sample space must sum to 1.
The probability of any event A is then obtained by summing the probabilities assigned to the simple events contained in A.
肚子穴位3. U NIONS AND INTERSECTION
The union of X, Y is formed by combining elements from each of the ts and is denoted Y X ∪.
Example 5 : X={1,4,7,9} and Y={2,3,4,5,6}
X or Y = Y X ∪= {1,2,3,4,5,6,7,9}
An element is included in Y X ∪if it’s in either X or Y or in both X and Y.
Y
X
好生意The interction
of X,Y contains the elements common to both ts and is denoted Y X ∩. Example 6 : X={1,4,7,9} and Y={2,3,4,5,6}
X and Y = Y X ∩= {4}
An element is included in Y X ∩if it’s in X and Y.
X
Y X ∩
Y
X
X={1,3,5,7,8,9}, Y={2,4,7,9} and Z={1,2,3,4,7}
a)Z X ∪=
1 4 9
7 2 3
5 6
b) Y X ∩=
c)Z X ∩=
d) Z Y X ∪∪=
e)Z Y X ∩∩=
f) Z )Y X (∩∪=
g))Y X ()Z Y (∩∪∩=
h) X or Y = Y X ∪=
i) Y and X = X Y ∩= Y X ∩=
Example 7.
A manufacturing plant conducted a survey to determine its employees’ reactions toward a propod change in working hours. A breakdown of the respons in the following table : Work area Reaction Total Agree Disagree
Production 17 23 40
Office 8 2 10
Total 25 25 50
Sample space
Suppo an employee is chon at random, with the relevant events being defined as follows:
A: the employee works in production.
B : the employee agrees with the propod change .
杨梅酒的功效和作用1. Express each of the following events in words:
a) A
b) (A or B)
c) (A and B)
d) A or B
2. Construct the contingency table with joint probabilities and marginal probabilities
3. Refer to question 1 and find the probabilities.
Answers :
1. a) A : The employee works in the office.
b) (A or B) : The employee either works in production or agrees with the propod change or both. c) (A and B) : The employee works in production and agrees with the propod change.
d) (A or B ) : The employee either works in production or disagrees with the propod
change or both.
2. The general form of a contingency table with probabilities is the following one :
P(A and B) = )B A (P ∩=5017=0.34 P(A and B) = 16.0508= P(A and B ) = 5023= 0.46 P(A and B ) =502= 0.04 P(A) = 8.05040=
P(A ) = 20.05010502508==
P(B) =5.05025=
P(B ) = 5
.05025
3. The probabilities are:
麦子地里的故事
a) P(A ) = 2.05010=
b) P(A or B) = ....)B A (P )B (P )A (P )B A (P =∩−+=∪
17=0.34
c) P(A and B) = )B
A(P∩=
50
d) P(A or B) = )B
A(P∪= P(A)+P(B)-P(A∩B)= …
4.C ONDITIONAL PROBABILITY
Let’s e the distinction between P(A) and P(A/B)
The (marginal) probability that an event A will occur is denoted by P(A).
P(A/B) is the conditional probability that A will occur given that the event B has occurred.
5.M UTUALLY EXCLUSIVE EVENTS AND INDEPENDENT EVENTS
Two events A and B are said to be independent if the knowledge that B has occurred does not affect the probability that A will occur. So, A and B are independent events if :
Equivalently, A and B are independent events if P(B/A) = P(B).感动中国十大人物事迹
The key to recognizing a conditional probability is to look for the words “given that” or their equivalent. For instance, the statement of a conditional probability might be read : “The probability that A will occur if B occurs”.
A and
B are mutually exclusive events if the occurrence of one event precludes the occurrence of the other event. A and B can’t occur simultaneously. So there is no interction between A and B.
⇒
Note: If A and B are mutually exclusive events, you can conclude that:
P(A∪B) = P(A) + P(B)
