Transcript Part IV - German Vargas

Part IV
Randomness and Probability
Chapter 14
From Randomness to Probability
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The Probability of an event is its long-run
relative frequency.
For any random phenomenon each
attempt, or trial, generates an outcome.
A combination of outcomes is called an
event
Two trials are independent if the
outcome of one trial doesn’t influence or
alter the probabilities of the other.
The Law of Large Numbers
(LLN)
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The long-run relative frequency of
repeated independent events gets
closer and closer to the true relative
frequency as the number of trials
increases.
NO law of averages!
Examples
Probability
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For any event A
0  P( A)  1
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The probability of the set of all possible
outcomes of a trial must be 1
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Ex: Coin : S={H,T}
Die : S={1,2,3,4,5,6}
The probability of an event occurring is 1
minus the probability that it doesn’t
occur
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S
S : Sample space represents the set of all
possible outcomes
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Venn Diagrams
P(A) = 1 – P(Ac)
Where AC is called the complement of A
Ac
A
Probability (cont.)
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Disjoint events
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Events that can’t occur
together
For two disjoint events
A and B, the probability
that one occurs is the
sum of the probabilities
of the two events
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P(A or B) = P(A) + P(B)
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(Provided that A and B are
disjoint)
B
A
Probability (cont.)
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For two independent
events A and B, the
probability that both A
and B occur is the
product of the
probabilities of the two
events
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P (A and B) = P(A) x P(B)
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(Provided that A and B are
independent)
B
A and B
A
Putting the Rules to work
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Coin
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Dice
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P(4 heads)=?
P(At least 1 head in 4 attempts)= ?
P(three 2’s)
P(4 in the addition of 2 dice)
Step-by-step page 374
Chapter 15
Probability Rules!
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When k possible outcomes are equally
likely, each has a probability of 1/k
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Ex : Coin, 1 die, pair of dice.
The General Addition Rule
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P(A or B) = P(A) + P(B) – P(A and B)
Ex: A:PC , B:Car
Exclusive or (XOR)
Contingency tables
Grades Popular Sports
Total
Boy
117
50
60
227
Girl
130
91
30
251
Total
247
141
90
478
Ex: P(Boy) = ?
P(Girl and Grades) = ?
(Cont.)
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Conditional Probability
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A probability that takes into account a given
condition .
P(B|A) = the Probability of B given A
P(B|A)= P(A and B)/P(A)
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Ex: Probability that a selected student wants to excel at
grades given that we have selected a girl
P(Grades|Girl)= P(Girl and Grades)/P(Girl)
Why P(A) cannot be zero?
The General Multiplication
Rule
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P(A and B) = P(A) x P(B|A)
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(doesn’t require independence)
Independence
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The outcome of one event does not influence the probability
of the other.
Events A and B are independent whenever
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P(grades|girl) = P(grades) ?
Disjoint events cannot be independent!
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(Symmetric)
Ex: Is the probability of having good grades as a goal
independent of the gender of the responding student.
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P(B|A)=P(B)
Ex: Grades
Step-by-step page 387 & 394
Drawing without replacement
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Ex : Poker
Ex : page 395
Tree Diagrams
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College Students (Page 396)
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44% Binge Drinking
37% Drink Moderately
19% Abstain Entirely
Tree Diagrams
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P(Binge and Accident)=
P(Binge)xP(Accident|Binge)
Reverse Conditioning
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P(Binge|Accident) = ???
Homework
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Project
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Find the probability of getting head or tail
for a coin toss, and the probability of
getting each number of the addition of a
pair of dice by recording and analyzing
several trials. (Hint: For the coin, figure
14.1 page 365. Hint for the pair of dice:
make a histogram)