Probability and Statistics Notes on Chapter 3 Probability

							              Probability and Statistics Notes on Chapter 3: Probability

                         Section 3.1: Basic Concepts of Probability
Uses of Probability: Probability affects decisions when the weather is forecast, when
marketing strategies are determined, insurance, investments, when medications are
selected, winning a lottery/raffle/bet, and when players are selected for professional
sports teams, etc. Probability is the basis of inferential statistics, which grew out of using
coins, dice and cards. These types of things were found in pyramids.
One abuse is to assume probabilities have “memories.” Throwing a coin has no memory
of what is landed on previously. Another common abuse is to incorrectly add
probabilities.

History: During the mid-1600s, a professional gambler made a lot of money on a
gambling game, he devised a new game and could not win so he contacted a
mathematician Blaise Pascal (1632-1692. Pascal became interested and began to study
probability, he corresponded with Pierre de Fermat (1601-1695), and the two formulated
the beginnings of probability theory. Pierre Laplace (1749-1827) studied probability and
is credited with putting probability on a sure mathematical footing.

The first book written on probability, The Book of Chance and Games, was written by
Jerome Cardan (1501 – 1575). Cardan was an astrologer, philosopher, physician,
mathematician, and gambler. This book contained techniques on how to cheat and how to
catch others at cheating.

Probability experiment: an action or trial through which specific results are obtained.
Example: Roll a die, flip a coin, or draw a card from a deck.

**Probability can be expressed as fractions (reduce), decimals (round to two or three
decimal places) or percentages. **

Outcome: result of single trial in a probability experiment.
Example: Roll a die, and get a 4. Flip a coin and get heads.

Sample space: set of all possible outcomes of a probably experiment.
Example: Roll a die and get a 1 or 2 or 3 or 4 or 5 or 6. Flip a coin and get either a Head
or Tail.

Event: consists of one or more outcomes and is a subset of the sample space.
Simple event: an event that consists of a single outcome, represented by uppercase letters
A, B, and C. Example: each person has only one type of blood.

                                     4 Rules of Probability
1. The probability of any event E is a number (either a fraction or a decimal) between and
including 0 and 1, 0 – 100%. This is denoted by 0  P(E)  1.
2. If an event cannot occur, then the probability is 0.
3. If an event is certain, then the probability is 1 (or 100%).


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4. The sum of the probability of the outcomes in the sample space is 1.

                           3 Basic Interpretations of Probability
1. Classical (theoretical) probability: each outcome in a sample space is equally likely
to occur.
                Number of outcomes in E                n( E )
P(E) =                                              =
        Total number of outcomes in sample space n( S )
Example: Choosing numbers for the lottery, rolling a die to land a 2, drawing a red king
out of a standard deck of cards.
                                                      Frequency of event E f
2. Empirical (experimental) probability: P(E) =                            
                                                         Total frequency      n
Example: Tossing a coin and getting head 28 times, 10 out of 20 students go to college.

3. Subjective probability: result from intuition, educated guesses, and estimates.
Example: There is a 30% probability that the patient needs an operation.

Law of large numbers: as you increase the number of times a probability experiment is
repeated, the empirical probability of an event approaches the theoretical probability of
an event. The more times you do the trials, the closer you get to the actual value.

Complement of event E: the set of all outcomes in a sample space that is not included in
event E. The complement of event E is denoted E  , read as E prime.
How do you know when to use this? Phrases- At least, none.
P(E’) = 1 – P(E)    or      P(E) = 1 – P(E)          or     P(E) + P(E’) = 1

               Section 3.2 Conditional Probability and Multiplication Rule

                                         P ( Aand B )
Conditional Probability: P( B A ) =
                                             P ( A)
The conditional probability of event B occurring, given that event A has occurred,
P( B A ) is read as probability of B, given A. The probability of an event occurring, given
that another event has already occurred. WHATEVER FOLLOWS THE WORD GIVEN,
GOES IN THE DENOMINATOR (bottom) OF THE FRACTION.
In a Venn Diagram it is the intersecting section of the two circles.

Independent : the occurrence of one of the events does not affect the probability of the
occurrence of the other event. To check if two events are independent: P( B )  P ( B A) .
Example: replace a card, tossing a coin and rolling a die are independent.

Dependent: events that are not independent. To check for independence: P ( B )  P ( B A) .
Example: do not replace a card, playing the piano makes you a better pianist

The Multiplication Rules for the Probability of A and B:
1. If events A and B are dependent, P(A and B) = P(A) P ( B A)


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2. If events A and B are independent, P(A and B) = P(A)  P(B).
Uses: in a sequence. Examples: 2 of something, playing cards and one is not replaced.

If you see the word and in a sentence: multiply. If you see the word or in a sentence: add.

AND = X                OR = + (look for overlap and subtract if necessary)

                              Section 3.3 The Addition Rule
Mutually exclusive: two events A and B cannot occur at the same time, have nothing in
common.
Example: Roll a die, getting an odd number and an even number.

Not mutually exclusive: two events A and B do have something in common.
Example: roll a die and get a 3, roll a die and get an odd number.

In discussing mutually exclusive and not mutually exclusive, Venn Diagrams are used.
John Venn (1834 – 1923) developed the diagrams and were used in set theory and
symbolic logic. The Venn diagrams have been adapted to probability also. The diagrams
only show a general picture of the probability rules, and do not portray all situations such
as P(A) = 0 accurately.

The Two Addition Rules for the Probability of A or B:
1. If two events are not mutually exclusive P(A or B) = P(A) + P(B) – P(A and B)
By subtracting P(A and B), you avoid double counting the probability of outcomes that
occur in both A and B.
2. If two events are mutually exclusive, P(A or B) = P(A) + P(B).

         Type of Probability                                Formula
Classical Probability                                  Frequency of outcomesin E
*Everyone has an equal chance             P(E)=
                                                Total number of outcomesin sample space
Empirical Probability                              Frequency of outcomes in E f
*Have different probabilities, not the    P(E) =                             
                                                        Total frequency        n
same for everyone
Complementary Events                      P(E) + P( E  ) = 1
*uses phrases such as at least, none      P(E) = 1 – P( E  )
                                          P( E  ) = 1 – P(E)
Multiplication Rule                       P(A and B) = P(A) P ( B A)
*Sequence- uses the word and              P(A and B) = P(A)  P(B) Independent
Addition Rule                             P(A or B) = P(A) + P(B) – P(A and B)
*Can not occur at same time- uses or      P(A or B) = P(A) + P(B) mutually exclusive




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                               Section 3.4 Counting Principles
The Fundamental Counting Principal: if one event can occur in m ways and a second
event can occur in n ways, the number of ways, the two events can occur in sequence is
m  n , also denoted k1  k2  k3... kn . Examples: outfits- 5 shirts, 3 shoes, 2 pants = 5*3*2
If there are two or more events in the problem, use FCP.

Factorial notation: denoted as n! Read as n factorial. In 1808 Christian Kramp first used
the factorial notation. n! = n(n – 1)(n – 2) . . . 1 3! = 3 x 2 x 1         0! = 1
Usually if there is only one number in the word problem, you use factorial.
Calculator: Type the value of n, Math, PRB, 4, Enter.

                         n!
Permutation: n Pr              an ordered arrangement of objects. (P.O.- order matters)
                      (n  r )!
The number of different permutations of n distinct objects is n!
Examples: combination lock, pin code, telephone number, trophies, SSN, zip code.
Calculator: type the value of n (bigger) number, Math; PRB; 2; Enter, type the value of r.
                                                                    n!
Distinguishable Permutations: of n objects where
                                                           n1 ! n2 !n3 !... nk !
                           n!
Combination: n Cr                  is a selection of r objects from a group of n objects
                       (n  r )!r !
without regard to order and is denoted by n Cr .
The number of combinations of r objects selected from a group of n objects.
Examples: committee, lottery, randomly select.
Calculator: Type the n number first, Math; PRB; 3; Enter, type the r number, Enter.


          Principle                          Description                   Formula
Fundamental Counting            If one event can occur in m ways and m  n ,
Principal                       a second event can occur in n ways,  k1  k2  k3... kn
                                the number of ways the two events
                                can occur in a sequence is m  n
Permutations Rule               The number of different ordered      n!
* ORDER MATTERS                 arrangements of n distinct objects.
                                The number of distinguishable                 n!
                                permutations of n objects.           n1 ! n2 !n3 !... nk !
                                The number of permutations of n                 n!
                                                                     nP 
                                                                        r
                                distinct objects taken r at a time.          (n  r )!
Combinations Rule               The number of combinations of r                   n!
                                                                     n Cr 
* ORDER DOES NOT                objects selected from a group of n           (n  r )!r !
MATTER                          objects without regard to order.




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