Transcript Chapter 5

5.1 Basic Probability Ideas
• Definition: Experiment – obtaining a piece of data
• Definition: Outcome – result of an experiment
• Definition: Sample space – list of all possible
outcomes of an experiment
• Definition: Event – collection of outcomes from
an experiment (a simple event is a single outcome)
• Definition: Equally likely sample space – sample
space in which all outcomes are equally likely
5.1 Basic Probability Ideas
• Definition: Probability – A number between 0
and 1 (inclusive) that indicates how likely an
event is to occur
• Definition: n(A) – number of simple events in A
• Definition: Theoretical probability of A – p(A)
= n(A)/n(S) = (# outcomes of A) ÷ (# outcomes
in the equally likely sample space)
5.1 Basic Probability Ideas
• Law of Large Numbers – as the number of
experiments increases without bound, the
proportion of a certain event approaches a
theoretical probability
• The Law of Large Numbers does not say: If you
throw a coin and get heads 10 times, that your
probability of getting tails increases. P(heads)
stays at 1/2
5.1 Basic Probability Ideas
• Empirical Probability – relative frequency
of an event based on past experience
p(A) =
(# times A has occurred) ÷ (# observations)
5.1 Basic Probability Ideas
•
Properties of Probabilities:
1. 0 ≤ p(A) ≤ 1
2. p(a) = 0  A is impossible, the event will
never happen
3. p(a) = 1  A is a certain event, the event
must happen
4. Let a1, a2, a3,…. an be all events in a sample
space, then p(a1) + p(a2) + … + p(an) = 1
5.2 Rules of Probability
•
•
Definition: Complement of an event A – The
event that A does not occur denoted AC
Properties:
1. P(A) + p(AC) = 1
2. P(AC) = 1 – p(A)
3. P(A) = 1 – p(AC)
•
Odds in favor of event A – n(A):n(AC) or
n(A) ÷ n(AC)
5.2 Rules of Probability
•
Example: p(A) = 1/3 then p(AC) = 1 – p(A) = 2/3
odds in favor of A = p(A):p(AC) = 1/3:2/3 = 1:2
•
odds against A = p(AC):p(A) = 2/3:1/3 = 2:1
Probabilities from odds in favor –
odds in favor = s:f (successes to failures)
p(A) = s/(s + f)
p(AC) = f/(s + f)
5.2 Rules of Probability
• Joint probability tables – displays possible
outcomes and their likelyhood of
occurrence
• Example: Given the following table of data:
Male
Coke
13
Pepsi
31
Female
22
14
5.2 Rules of Probability
• Probability table for example (total of 80
people in the sample):
Male
Coke
Pepsi
13/80 = .1625 31/80 = .3875
Female
22/80 = .275
14/80 = .175
5.2 Rules of Probability
•
Simple probability tree:
Boy
branches
root
Girl
5.2 Rules of Probability
•
•
Probability trees are useful when events
do not have the same probability (there is
no equally likely sample space)
Problem solutions involving trees can
become long if many branches are to be
calculated (similar to the brute force
method in section 4.5)
5.3 Probabilities of Unions and Intersections
• Definition: The union of two events A and B is the event
that occurs if either A or B or both occur in a single
experiment. The union of A and B is denoted A  B
Example: (rolling a die – getting an even number or a
perfect square)
2
4
6
1
3
5
5.3 Probabilities of Unions and Intersections
• Definition: The intersection of two events A and B is the
event that occurs if both A and B occur in a single
experiment. The intersection of A and B is denoted A  B
Example: (rolling a die – getting an even number and a
perfect square)
2
4
6
1
3
5
5.3 Probabilities of Unions and Intersections
•
Definition: mutually exclusive or disjoint events
– events for which A  B =  (where 
represents an event with no elements)
• If A and B are mutually exclusive, then:
1. P(A  B) = 0
2. P(A  B) = P(A) + P(B)
• Union Principle of Probability:
P(A  B) = P(A) + P(B) - P(A  B)
5.4 Conditional Probability and
Independence
• Definition: The conditional probability of A given B is the
probability of A occurring given that B has already
occurred – denoted P(AB)
When outcomes are equally likely:
n(AB)
P(AB) = n(B)
• Conditional Probability Formula (outcomes not necessarily
equally likely)
P(AB)
P(AB) = P(B)
5.4 Conditional Probability and
Independence
• Multiplication Principal:
P(A  B) = P(B)  P(AB)
• Tree diagrams – useful for conditional
probability because each section of a branch
is a probability conditional by the previous
branches
5.4 Conditional Probability and
Independence
• Independence: Two events A and B are said to be
independent if the occurrence of A does not affect P(B) and
vice versa.
A & B are independent if:
P(AB) = P(A) or P(BA) = P(B)
• Multiplication Principle for Independent Events:
A & B are independent events
 P(A  B) = P(A)  P(B)
5.5 Bayes’ Formula
• Bayes formula for 2 cases:
P(AB) =
P(A)  P(BA)
P(A)  P(BA) + P(AC)  P(BAC)
5.5 Bayes’ Formula
• Bayes formula for n disjoint events:
P(AiB) =
P(Ai)  P(BAi)
P(A1)  P(BA1) + P(A2)  P(BA2) + … + P(An)  P(BAn)
5.6 Permutations and Combinations
• Multiplication Principle – given a tree with the
number of choices at each branch being m1, m2,
m3, … mn, then the number of possible
occurrences is:
m1  m2  m3  …  mn
5.6 Permutations and Combinations
• Permutations: The number of arrangements of r
items from a set of n items.
Note: Order matters.
nP r
=
n!
(n – r)!
5.6 Permutations and Combinations
• Combinations: The number of subsets of r items
from a set of n items.
Note: Order does not matter.
nCr
=
n!
(n – r)!  r!
5.6 Permutations and Combinations
- summary of counting formulas
With replacement
(order matters)
Without replacement
Order matters
(arrangements)
Multiplication
principal
Permutation
Order does
not matter
(subsets)
Combination
5.7 Probability and Counting Formulas
•
Example: A bag contains 4 red marbles and 3
blue marbles. Find the probability of selecting:
a. Two red marbles
b. Two blue marbles
c. A red marble followed by a blue marble
# ways to pick 2 red
= 4C2 = 6/21 = 2/7
P(2 red) =
# ways to pick any 2 marbles
7C2
5.7 Probability and Counting Formulas
# ways to pick 2 blue
P(2 blue) = # ways to pick any 2 marbles
= 3C2
7C2
P(red then blue) =
= 3/21 = 1/7
chance of picking red on first 
chance of picking blue on second
= 4/7  3/6 = 2/7
5.7 Probability and Counting Formulas
• Birthday Problem: Suppose there are n people in a room.
Find the formula for the probability that at least two people
have the same birthday.
Note: P(at least two birthdays the same)
= 1 – P(no two birthdays the same)
# ways for n people to have birthdays = 365n
# ways for for n birthdays without repeats = 365Pn
answer = 1 – (365Pn  365n)
5.8 Expected Value
• Expected Value – for a given sample space with
disjoint outcomes having probabilities p1, p2, p3,
… pn and a value (winnings) of x1, x2, x3, … xn ,
then the expected value of the sample space is:
x1 p1 + x2 p2 + x3 p3 +…. xn pn
• Definition: A game is said to be fair if the cost of
participating equals the expected winnings.
– Expected winnings < cost  unfair to participant
– Expected winnings > cost  unfair to organizers
5.9 Binomial Experiments
•
Definition: Binomial Experiment
1. The same trial is repeated n times.
2. There are only 2 possible outcomes for each
trial – success or failure
3. The trials are independent so the probability
of success or failure is the same for each trial.
5.9 Binomial Experiments
•
Binomial Probabilities:
P(x successes) = nCr  px  (1-p)n-x with x = 0,1,2,…,n
where n is the number of trials, p is the probability of
success, and x is the number of successful trials
•
Expected value of a binomial experiment = n  p
note: The most likely outcome “tends to be close to” the
expected value.