WHAT IS PROBABILITY

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WHAT IS PROBABILITY?

• prof. Renzo Nicolini I.M.G.CARDUCCI - Trieste

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WHAT IS PROBABILITY?

• CLIL project
• Class II C

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LESSON 1

• CLIL project
• Class II C

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WHAT IS PROBABILITY?

• The word probability
• derives
• from the Latin probare

(to prove, or to test).
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WHAT IS PROBABILITY?
PROBABLE is almost synonym of

• likely
• hazardous
• risky
• uncertain
• doubtful

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WHAT IS PROBABILITY?
theory of probability attempts to quantify the
notion of probable.
HOW PROBABLE SOMETHING IS?
/LIKELY
To answer, we need a number!!!!!
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HISTORICAL REMARKS

• The scientific study of probability is a modern
  development.
• Gambling
• shows that there has been an interest in
  quantifying the ideas of probability for
  millennia,
• but exact mathematical descriptions of use in
  those problems only arose much later.

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HISTORICAL REMARKS

• The doctrine of probabilities starts with the
  works of
• Pierre de Fermat
• Blaise Pascal (1654)
• Christian Huygens(1657)
• Daniel Bernoulli (1713)
• Abraham de Moivre (1718)

Blaise Pascal
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• Vocabulary

An experiment
is a situation involving chance or probability
that leads to results called outcomes.
An outcome
is the result of a single trial of an experiment.
An event
is one or more outcomes of an experiment.
Probability
is the measure of how likely an event is.
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Experiments!
EXPERIMENT OUTCOME EVENT

• Rolling a single 6-sided die
• Running an horse race
• Driving a car race
• Picking a card from a deck
• Tossing a coin.

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Outcomes!
EXPERIMENT OUTCOME EVENT

• Rolling a single 6-sided die
• a number six was drawn
• (to be drawn uscire)
• a number three was drawn
• a number eight cant be drawn

Possible outcomes in the experiment
Impossible outcome in the experiment
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Outcomes!
EXPERIMENT OUTCOME EVENT

• Experiment Driving a car race
• Outcome Schumacher wins

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Outcomes!
EXPERIMENT OUTCOME EVENT

• Experiment Picking a card from a deck
• Outcome A king is drawn

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Outcomes!
EXPERIMENT OUTCOME EVENT

• Experiment Tossing a coin
• Outcome a tail has been tossed

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Outcomes!
EXPERIMENT OUTCOME EVENT

• THE SET OF ALL THE POSSIBLE OUTCOMES IS CALLED
  SAMPLE SPACE and is denoted by S.

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Outcomes! Examples
EXPERIMENT OUTCOME EVENT

• Experiment Rolling a die once
• Sample space S 1,2,3,4,5,6
• Experiment Tossing a coin
• Sample space S Heads,Tails
• Experiment Measuring the height (cms) of a girl
  on her first day at school
• Sample space S the set of all (?) possible real
  numbers

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Event!
EXPERIMENT OUTCOME EVENT

• Its the particular outcome or set of outcomes
  Im interested to study
• How possible is that a Queen is picked up from
  a deck of cards?
• How possible is that a Jack OR a King are
  picked up from a deck of card?
• Rolling a die once, how possible is it that the
  score is lt 4?

This is OUR event!
This is OUR event!
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Probability!

• We call probability the value we estimate for a
  single event
• What is the probability that a Queen is picked
  up from a deck of card?
• What is the probability that a Jack OR a King
  is picked up from a deck of card?

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LETS REPEAT!!
WHEN I DO SOMETHING I SAY THAT I CARRY OUT AN
EXPERIMENT
EXAMPLES?
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LETS REPEAT!!
ANY POSSIBLE SITUATION THAT OCCURS WHEN I CARRY
OUT THE EXPERIMENT IS AN
OUTCOME
EXAMPLES?
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LETS REPEAT!!
ALL THE POSSIBLE OUTCOMES THAT CAN OCCUR WHEN I
EXECUTE THE EXPERIMENT, FORM THE
SAMPLE SPACE
EXAMPLES?
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LETS REPEAT!!
THE PARTICOLAR OUTCOME or SET OF OUTCOMES WERE
INTERESTED IN IS AN
EVENT
EXAMPLES?
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LETS REPEAT!!
THE MEASURE OF HOW LIKELY AN EVENT IS, IS CALLED
PROBABILITY
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WHAT IS PROBABILITY?

• CLIL project
• Class II C

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LESSON 2

• CLIL project
• Class II C

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HOW TO EVALUATE PROBABILITY?
Probability is a number!
We need a formula or a procedure to find it!
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HOW TO EVALUATE PROBABILITY?
THERE ARE THREE POSSIBLE WAYS TO FIND THIS VALUE

• CLASSICAL DEFINITION
• SUBJECTIVE PROBABILITY
• FREQUENTIST DEFINITION

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HOW TO EVALUATE PROBABILITY?
A CURIOSITY!!

• SUBJECTIVE PROBABILITY was proposed in XX
  century by Bruno De Finetti, who worked in Triest
  (Generali, University) from 1931 to 1954

We will not talk about this type of probability
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HOW TO EVALUATE PROBABILITY?
WELL SEE ONLY THE CLASSICAL DEFINITION OF
PROBABILITY.
by SIMON DE LAPLACE (1749-1827)
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CLASSICAL PROBABILITY

• SIMON DE LAPLACE (1749-1827) gave the most
  famous definition of probability.
• Its called
• CLASSICAL DEFINITION OF PROBABILITY

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Mathematics need fomulas!

• In order to measure probabilities, he has
  proposed the following formula for finding the
  probability of an event.

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THE FORMULA FOR THE CLASSICAL PROBABILITY

• Probability Of An Event P(A)
•    The Number Of Ways an Event A Can Occur 
• The Total Number Of Possible Outcomes

The number of elements of the sample space
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THE FORMULA FOR THE CLASSICAL PROBABILITY

• The probability of event A is the number of
  ways event A can occur divided by the total
  number of possible outcomes.

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THE FORMULA FOR THE CLASSICAL PROBABILITY

• The probability of event A is the number of
  favorable cases (outcomes) divided by the total
  number of possible cases (outcomes).

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EXAMPLE/EXERCISE
A single 6-sided die is rolled.

• What is the probability of each outcome?
• What is the probability of rolling an even
  number?
• Of rolling an odd number?

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Outcomes  The possible outcomes of this
experiment are 1, 2, 3, 4, 5 and 6.
P(1)    number of ways to roll a 1    1
total number of sides 6   P(2)
   number of ways to roll a 2    1
total number of sides 6   P(3)   
number of ways to roll a 3    1
total number of sides 6   P(4)   
number of ways to roll a 4    1
total number of sides 6   P(5)   
number of ways to roll a 5    1
total number of sides 6   P(6)   
number of ways to roll a 6    1
total number of sides 6
All the values are the same!!! The outcomes are
equally likely .
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EQUALLY LIKELY EVENTS
EQUALLY LIKELY EVENTS HAVE THE SAME PROBABILITY
TO OCCUR
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What is the probability of rolling an even
number?

• P(even)    ways to roll an even number  
  total number of sides
   

probability of rolling an even number is one
half 0,5
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What is the probability of rolling an odd number?
probability of rolling an odd number is
one half
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NOTE classical probability is a priori
Its interesting to note that, in order to
calculate the probability in the classical way,
its necessary to know EVERYTHING about the
experiment.

• We need to know the possible outcomes (the whole
  sample space),
• we need to know the EVENT we are interest in.
• In few words, WE HAVE TO KNOW EVERYTHING BEFORE
  RESULTS COME OUT.
• Thats why we say that
• CLASSICAL PROBABILITY IS
• A PROBABILITY A PRIORI.

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Probability Of An Event P(A)    The Number
Of Ways Event A Can Occur  The Total Number Of
Possible Outcomes
Some more about the formula for probability

• The impossible event
• The certain event

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Is it possible that there are no ways event A
can occur?
SURE!
In this case the formula for probability  The
Number Of Ways Event A Can Occur  The Total
Number Of Possible Outcomes has numerator equal
to 0!
P(A) 0
THE EVENT IS IMPOSSIBLE!
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Is it possible that there are no ways event A
can occur?
THE EVENT IS IMPOSSIBLE?
P(A) 0
It has no probability to happen!
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EXAMPLE OF PROBABILITY 0
Which is the probability of rolling number 7 on a
6 sided die?
The Number Of Ways Event A Can Occur  The Total
Number Of Possible Outcomes
The Number Of Ways Event A Can Occur is 0
because number 7 doesnt exist in such a die!!!
P(A) 0
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Is it possible that event A certainly will
occur?
SURE!
In this case the formula for probability  The
Number Of Ways Event A Can Occur  The Total
Number Of Possible Outcomes has numerator equal
to the denominator.
The fraction values 1
THE EVENT IS CERTAIN!
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Is it possible that event A certainly will
occur?
WHICH IS THE PROBABILITY THAT, ROLLING A DIE, A
NUMBER BETWEEN 0 AND 7 COMES OUT ?
P(A)  The Number Of Ways an Event A Can Occur
6 1 The Total Number Of Possible
Outcomes 6
P(A) 1
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Is it possible that event A certainly will
occur?
THE EVENT IS CERTAIN!
P(A) 1
It will certainly happen!
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LETS REPEAT!!
PROBABILITY

• is a positive real number, between 0 and 1

Zero for the impossible event
One for the certain event
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LETS REPEAT!!
TO FIND THE CLASSICAL PROBABILITY (Laplace) we
need the following definition  P(A)  
The Number Of Favorable Cases The
Total Number Of Possible Cases
The number of elements of the sample space
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LETS REPEAT!!
WHEN TWO EVENTS HAVE THE SAME PROBABILITY, WE SAY
THAT THEY ARE EQUALLY LIKELY.
Heads and tails are equally likely!!!
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LAST QUESTION!

• Which is the probability that next time youll
  appreciate our CLIL lesson?

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LAST ANSWER!

• Classical probability doesnt give any answer
  to this question, because its not a problem
  solving a priori.

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LAST ANSWER!

• Its a typical situation of subjective
  probability, which depends on your particular
  feeling about the event We come next time. Its
  a result of you own sensation!!

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LAST ANSWER!

• I HOPE THIS PROBABILITY IS NOT
• ZERO!

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WHAT IS PROBABILITY?

• CLIL project
• Class II C

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LESSON 3

• CLIL project
• Class II C

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LETS REPEAT!!
WHEN I DO SOMETHING I SAY THAT I EXECUTE AN
EXPERIMENT
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LETS REPEAT!!
ANY POSSIBLE SITUATION THAT OCCURS WHEN I EXECUTE
THE EXPERIMENT IS AN
OUTCOME
ALL THE POSSIBLE OUTCOMES THAT CAN OCCUR WHEN I
EXECUTE THE EXPERIMENT, FORM THE
SAMPLE SPACE
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LETS REPEAT!!
THE PARTICOLAR OUTCOME or SET OF OUTCOMES WERE
INTERESTED IN, IS AN
EVENT
THE MEASURE OF HOW LIKELY AN EVENT IS, IS CALLED
PROBABILITY
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THE FORMULA FOR THE CLASSICAL PROBABILITY

• Probability Of An Event P(A)
•    The Number Of Ways an Event A Can Occur 
• The Total Number Of Possible Outcomes

The number of elements of the sample space
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THE FORMULA FOR THE CLASSICAL PROBABILITY

• The probability of event A is the number of
  favorable cases (outcomes) divided by the total
  number of possible cases (outcomes).

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• SOMETHING MORE ABOUT THE THEORY OF PROBABILITY

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PROBLEM

• Imagine to be asked to solve the following
  exercise

Rolling a die, which is the probability of
rolling any number except 2?
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PROBLEM Rolling a die, which is the probability
of rolling any number except 2.
The statement any number except number 2 is
the negation of the statement number 2
The EVENT A any number except 2 is the negation
of the EVENT rolling a number 2
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PROBLEM Rolling a die, which is the probability
of rolling any number except 2?
We say that the EVENT any number except number
2 is the COMPLEMENT OF THE EVENT A rolling a
number 2
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COMPLEMENT OF AN EVENT A

• Its the opposite statement of the EVENT A

We use to indicate it with A
A (A bar) is the complement of A
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Rolling a die, which is the probability of
rolling any number except 2?
Lets calculate the probability of A
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PROBLEM Rolling a die, which is the probability
of rolling any number except 2?

• What do we need?

Favorable cases.
All the elements of subset ?1,3,4,5,6 ?
Possible cases. ?1,2,3,4,5,6 ?
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PROBLEM Rolling a die, which is the probability
of rolling any number except 2?
The probability of this event is 5 6
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PROBLEM Rolling a die, which is the probability
of rolling any number except 2?
Can we solve this problem in another way?
YES!
HOW?
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PROBLEM Rolling a die, which is the probability
of rolling a 2?
Lets start considering the problem of the EVENT
A Probability of rolling a 2

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PROBLEM Rolling a die, which is the probability
of rolling a 2?
ITS OBVIOUSLY 1 6

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PROBLEM Rolling a die, which is the probability
of rolling WHICHEVER number BUT A 2?
Now, we have two data
Probability one-sixth for EVENT A
Probability five-sixth for EVENT A
OBSERVE THAT ONE-SIXTH PLUS FIVE-SIXTH IS EQUAL
TO
1
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PROBABILITY OF THE COMPLEMENT

• The probability we found is 1.

In other words, the sum between P(A) and P (A)
is 1.
P(A) P (A) 1
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PROBABILITY OF THE COMPLEMENT
P(A) P (A) 1
IS IT A FORTUITOUS CASE?
NO! Its a general rule!!!!
LETS PROVE IT!!!!!
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P(A) P (A) 1

• Let n(A) be the number of favorable cases for the
  event A
• Let n(A) be the number of favorable cases for the
  event A
• Let n(E) be all the number of all possible cases
  for the experiment (the number of elements in the
  sample space

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P(A) P (A) 1

• Its quite obvious that
• n(A) n(A) n(E)

In fact, the number of cases for A and the number
of cases for A exhaust all the possibilities,
i.e. all the possible cases.
Lets divide the equality by n(E)
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P(A) P (A) 1
and than, with an obvious (?) passage,
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PROBABILITY OF THE COMPLEMENT
WE GOT THE RESULT P(A) P (A) 1
OR, IN OTHER TERMS, The probability P (A) of the
complement of an event A is given by the
subtracting from 1 the probability P (A) of the
event A
P (A) 1- P(A)
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PROBABILITY OF THE COMPLEMENT
In general, if we know the probability of an
event, we can immediately calculate the
probability of its complement !
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PROBABILITY OF THE COMPLEMENT
A single card is chosen from a standard deck of
52 cards. What is the probability of choosing a
card that is not a King?
P (not a king) 1 P(king)
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LETS REPEAT!!
When an event B has the opposite requirements of
an event A, we say that the event B is the
COMPLEMENT OF EVENT A
We can also indicate it as A
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LETS REPEAT!!
We have proved that the following formula for
finding the value of the complement of an event
P(A) P (A) 1
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WHAT IS PROBABILITY?

• CLIL project
• Class II C

85
LESSON 4

• CLIL project
• Class II C

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LETS REPEAT!!
We have proved the following formula for the
complement
P(A) P (A) 1
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LETS REPEAT!! PROBABILITY OF THE COMPLEMENT
A single card is chosen from a standard deck of
52 cards. What is the probability of choosing a
card that is not a King?
P (not a king) 1 P(king)
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COMPOUND EVENT
Lets see now a more complicated situation,
involving more actions.
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Experiment.

• Lets suppose to propose an experiment in
  which we do two actions.

AND
1. We roll a die
2. We take a number, playing tombola
and
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Experiment.

• we set up an event which includes both actions

For instance What is the probability of drawing
an odd number from the sack of tombola AND
rolling a multiple of 3 on the die?
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INDEPENDENT EVENTS

• First of all, we can notice that the rolling
  of the die and the drawing of the number are
• INDEPENDENT EVENTS.

Two events, A and B, are independent if the fact
that A occurs does not affect the probability of
B occurring.
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INDEPENDENT EVENTS

• The rolling of the die and the drawing of the
  number playing tombola are
• INDEPENDENT EVENTS

because the tombola number doesnt see the
outcome of the die and it isnt influenced by it
!!!!!
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Experiment.
What is the probability of drawing an odd number
from the sack of tombola AND rolling a multiple
of 3 on the die?
We are looking for the probability of a more
complicated event which involves two simpler
INDEPENDENT events
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COMPOUND EVENT
TWO ACTIONS
ONE REQUIREMENT
CONNECTOR AND
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COMPOUND EVENT
This is a typical example of COMPOUND EVENT
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COMPOUND EVENT
In the COMPOUND EVENT we want that both the
events occur
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COMPOUND EVENT
In the COMPOUND EVENT we want that one AND the
other event occur
AND
The KEY CONJUNCTION is
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COMPOUND EVENT
We could prove that the probability of the
compound event is always the product of the
single probabilities of two independent events
which compose the compound event.
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Multiplication rule

• When two events, A and B, are independent,
  the probability of both occurring is

 P(A and B) P(A) P(B)
This is called the multiplication rule
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Experiment.
What is the probability of choosing an odd number
from the sack of tombola AND rolling a multiple
of 3 on the die?
EVENT A drawing an odd tombola number
P(odd)
EVENT B rolling a multiple of 3
P(3,6)
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Experiment.
What is the probability of drawing an odd number
from the sack of tombola AND rolling a multiple
of 3 on the die.
We can use the multiplication rule because the
two events are evidently independent!
P(odd AND die) P(odd) P(die)
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OTHER EXPERIMENTS
A coin is tossed and a single 6-sided die is
rolled. Find the probability of tossing heads AND
rolling a 3 on the die.
P(head)
P(head AND 3)
P(3 on die)
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OTHER EXPERIMENTS

• A card is chosen at random from a deck of 52
  cards. It is then put back and a second card is
  chosen. What is the probability of choosing a
  jack AND an eight, replacing the chosen card?

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OTHER EXPERIMENTS. A card is chosen at random
from a deck of 52 cards. It is then put back and
a second card is chosen. What is the probability
of getting a jack AND an eight, replacing the
chosen card?

P(jack)
P(eigth)
P(jack AND eight)
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DEPENDENT EVENTS

• What happens if we decide not to put back the
  first card in the deck?

In this case, the second draw would be
conditioned by the first one. In fact, in the
second draw there would be only 51 cards in the
deck! So, in the second draw, the possible cases
would be 51 (one card has been removed!), while
the favorable cases of picking an eight would
remain the same (4).
P(eigth)
P(jack)
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DEPENDENT EVENTS

• What happens if we decide not to put back the
  first card in the deck?

In this case, the second draw would be
conditioned by the first one. In fact, in the
second draw there would be only 51 cards in the
deck! So, in the second draw, the possible cases
would be 51 (one card has been removed!), while
the favorable cases of picking an eight would
remain the same (4).
THE EVENTS ARENT INDEPENDENT ANYMORE. THEY ARE
DEPENDENT.
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DEPENDENT EVENTS
P(eigth)
P(jack)
P(jack AND eight)
PROBABILITY HAS CHANGED!
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AN OTHER EXPERIMENT

• A jar contains 3 red, 5 green, 2 blue and 6
  yellow marbles. A marble is picked at random from
  the jar. After putting it back, a second marble
  is picked. What is the probability of getting a
  green and a yellow marble?

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AN OTHER EXPERIMENT
Possible cases 16 with replacing
P(yellow)
P(green)
P(yellow AND green)
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DEPENDENT EVENTS

• But what happens if we decide not to put back the
  first marble in the jar ?

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ANOTHER EXPERIMENT 3 red, 5 green, 2 blue and 6
yellow
Possible cases 16 without replacing
P(yellow)
P(green)
P(yellow AND green)
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ANOTHER EXPERIMENT 3 red, 5 green, 2 blue and 6
yellow
with replacing
P(yellow AND green)
with no replacing
P(yellow AND green)
PROBABILITY HAS CHANGED AGAIN!
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COMPOUND EVENT

• If the events are independent, multiplication
  rule is valid and probability is just
• If the events are dependent, multiplication rule
  is still valid, but the second factor depends on
  the first

 P(A and B) P(A) P(B)
 P(A and B) P(A) P(B A)
Read B occurs given that event A has occurred
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LETS REPEAT the multiplication rule

• When two events, A and B, are independent,
• the probability of both occurring is

 P(A and B) P(A) P(B)
When two events, A and B, are dependent, the
probability of both occurring is
 P(A and B) P(A) P(B A)
The usual notation for "event B occurs given that
event A has occurred" is B A" (B given A).
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WHAT IS PROBABILITY?

• CLIL project
• Class II C

116
LESSON 5

• CLIL project
• Class II C

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LETS REPEAT!! Independent events
Two events, A and B, are independent if the fact
that A occurs does not affect the probability of
B occurring.
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LETS REPEAT!!
The rolling of a die and the drawing of a tombola
number are INDEPENDENT events!
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LETS REPEAT!!
When we have two events and we want that both of
those occur, we are considering a COMPOUND EVENT
120
LETS REPEAT!!
In the COMPOUND EVENT we want that one AND the
other event occur
AND
The KEY CONJUNCTION is
121
LETS REPEAT!!
We could prove that the probability of the
compound event is always the product of the
single probabilities of two independent events
which form the compound event.
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LETS REPEAT!!

• When two events, A and B, are independent,
  the probability of both occurring is

 P(A and B) P(A) P(B)
This is called the multiplication rule
123
LETS REPEAT!!

• A card is chosen at random from a deck of 52
  cards. It is then put back and a second card is
  chosen. What is the probability of drawing a jack
  AND an eight, putting back the chosen card?

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DEPENDENT EVENTS
In this case we say that we are valuating A
COMPOUND EVENT of TWO DEPENDENT EVENTS
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LETS REPEAT!!! A card is chosen at random from a
deck of 52 cards. It is then replaced and a
second card is chosen. What is the probability of
choosing a jack AND an eight, replacing the
chosen card?

P(jack)
P(eigth)
P(jack AND eight)
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LETS REPEAT the multiplication rule

• When two events, A and B, are independent,
• the probability of both occurring is

 P(A and B) P(A) P(B)
When two events, A and B, are dependent, the
probability of both occurring is
 P(A and B) P(A) P(B A)
The usual notation for "event B occurs given that
event A has occurred" is B A" (B given A).
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MORE FAVORABLE OUTCOMES
Lets see know a new situation, involving only
one experiment and one action (one rolling of the
die, one choosing of a card, and so on), but
where we accept
MORE FAVORABLE OUTCOMES
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Experiment. MORE FAVORABLE OUTCOMES
We choose a card from an ordinary deck, and we
accept
a numbered card.
OR
either a King
first favorable event
second favorable event
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MUTUALLY EXCLUSIVE EVENTS

• First of all, we can notice that the choosing
  of a King and the choosing of a numbered card
  CANT OCCURE at the same time. We say that
• They are two
• MUTUALLY EXCLUSIVE EVENTS
• (disjoint events)

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MUTUALLY EXCLUSIVE EVENTS
Two events are mutually exclusive (or disjoint)
if it is impossible for them to occur together.
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Experiment.
We are looking for the probability of a more
complicated event which involves two simpler
MUTUALLY EXCLUSIVE events.
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ONE ACTION
TWO REQUESTS
CONNECTOR OR
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COMPOUND EVENT
We could prove that the probability of the
occurring of one of two MUTUALLY EXCLUSIVE events
is the sum of the probabilities of each event.
134
Addition rule

• When two events, A and B, are mutually
  exclusive, the probability of just one occurring
  is

 P(A or B) P(A) P(B)
This is called the addition rule
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ANOTHER EXPERIMENT

• A jar contains 3 red, 5 green, 2 blue and 6
  yellow marbles. A marble is chosen at random from
  the jar. What is the probability of choosing a
  green OR a yellow marble?

136
ANOTHER EXPERIMENT
Possible cases 16
P(yellow)
P(green)
second favorable event
first favorable event
P(yellow OR green)
137
ONE MORE EXERCISE

• What is the probability of rolling a 2 OR a 5
  on a single 6-sided die?

MUTUALLY EXCLUSIVE EVENTS
 P(A ? B) P(A) P(B)
Read OR
P(A or B)
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MUTUALLY EXCLUSIVE EVENTS

• Formally, if two events A and B are mutually
  exclusive we can write
• A ? B ?
•         

WHY? WHAT DOES IT MEAN?
139
A ? B ?
Outcomes which satisfy elementary event A
sample space
A
B
There arent outcomes which contemporarily
satisfy event A and event B
Outcomes which satisfy elementary event B
A ? B ? ? Disjoint sets
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A ? B ? In a class

• Event A randomly, choosing a green-eyes-student
  
• Event B randomly, choosing a brown-eyes-student
  

141
A ? B ? In a class
A green-eyes-student
B brown-eyes-student
A
B
Obviously, no student has contemporarily green
and brown eyes
142
A ? B ? In a class
Obviously, no student has contemporarily green
AND brown eyes
Therefore, the probability of choosing an element
of A and an element of B is 0
The sets are disjoint
The events are disjoint
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LETS REPEAT the addition rule

• When two events cant occur contemporarily, they
  are said

- MUTUALLY INDEPENDENT EVENTS
- DISJOINT EVENTS
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LETS REPEAT the addition rule

• When two events cant occur contemporarily,
  theres no element in the intersection between A
  and B

A ? B ?
 P(A ? B) P(A) P(B)
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WHAT IS PROBABILITY?

• CLIL project
• Class II C

146
LESSON 6

• CLIL project
• Class II C

147
LETS REPEAT the addition rule

• When two events cant occur contemporarily, they
  are said

- MUTUALLY INDEPENDENT EVENTS
- DISJOINT EVENTS
148
LETS REPEAT the addition rule

• When two events cant occur contemporarily,
  theres no element in the intersection between A
  and B

A ? B ?
 P(A ? B) P(A) P(B)
149
Addition rule for NOT MUTUALLY EXCLUSIVE EVENTS
BUT WHAT ABOUT WHEN TWO EVENTS CAN OCCUR
CONTEMPORARILY?
i.e. WHAT IS THE PROBABILITY OF TWO NOT MUTUALLY
EXCLUSIVE EVENTS?
LETS SEE AN EXAMPLE!
150
OTHER EXPERIMENTS
A card is chosen at random from a deck of 52
cards.
What is the probability of choosing a jack OR a
club? ?
151
ONE ACTION (one picking of a card)
TWO REQUESTS (club jack)
CONNECTOR OR
152
NOT MUTUALLY EXCLUSIVE EVENTS

• In this case the event A (choosing a jack)
  and the event B (choosing a club? ) are not
  disjoint.

153
Outcomes which satisfy elementary event A 4 jacks
In this case the event A (choosing a jack)
and the event B (choosing a club) are not
disjoint.
sample space 52 cards
Outcomes which satisfy both the events the jack
of clubs
7
2
3
J
J
J
Q
8
5
10
J
K
4
1
9
6
THE EVENTS HAVE AN INTERSECTION
THEY ARE NOT DISJOINT
THEY ARE NOT MUTUALLY EXCLUSIVE
HOW TO EVALUATE THE PROBABILITY IN THIS CASE OF
NOT MUTUALLY EXCLUSIVE EVENTS?
Outcomes which satisfy elementary event B
13 clubs.
154
How to evaluate the probability in this case of
not mutually exclusive events?
How many outcomes are possible ( form the sample
space)?
52 (number of cards)
155
How to evaluate the number of favorable cases
when events are not mutually exclusive?
How many outcomes are favorable?
4 (number of jacks)

Is it correct?
NO!
13 (number of clubs ? )
17 (number of jacks OR clubs)
156
How to evaluate the number of favorable cases
when events are not mutually exclusive?
sample space 52 cards
Outcome which satisfies both the events the
jack of clubs
4 jacks
7
9
K
7
7
2
2
4
3
J
J
J
Q
8
5
5
10
7
J
K
4
10
4
1
9
10
6
1
13 clubs ?
Q
6
1
3
3
2
3
9
The real number of favorable cases is 16, and
not 17 we have counted the jack of clubs
twice!!!!!!!
157
How to evaluate the number of favorable cases
when events are not mutually exclusive?
The correct procedure to find the number of
favorable cases is
n(fav.cases)
n(jacks)
n(clubs)
? n(jacks ? clubs)
4
13
? 1
16
We must subtract the number of the elements of
the intersection not to count them twice!
158

• So, the probability of choosing a jack OR a club
  ? is

But lets write the first fraction in another way
159
P(jack)
OR
P(clubs)
P(jack?clubs)
AND
160
GENERAL FORMULA

• We can generalize what we have just found

P(jack?clubs) P(jacks) P(clubs) ? P(jack?clubs)
P(A?B) P(A) P(B) ? P(A?B)
161
AN OTHER EXPERIMENT

• Playing tombola, what is the probability that
  the first extracted number is
• MULTIPLE OF 10 (A) OR GREATER THAN 70 (B)?

The events A and B are not disjont!
We will use formula for not mutually exclusive
events
P(A?B) P(A) P(B) ? P(A?B)
162
Playing tombola, what is the probability that the
first extracted number is MULTIPLE OF 10 (A) OR
GREATER THAN 70 (B)?
9
?10,20,30,40,90?
P(A)
20
?71,72,73,89,90?
P(B)
2
?80,90?
P(A?B)
163
LETS REVIEW the addition rule

• When two events cant occur contemporarily,
  theres no element at the intersection between A
  and B

A ? B ?
 P(A ? B) P(A) P(B)
164
LETS REVIEW the addition rule

• When two events can occur contemporarily, theres
  some element at the intersection between A and B

A ? B ? ?
 P(A ? B) P(A) P(B) - P(A ? B)