IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
Probability Theory
1. Deterministic Experiments
With
deterministic
experiments ,
the
results
can
be
predicted before the experiment is conducted.
Example
If a stone is dropped from a window, it is know n, undoubtedly,
that the stone will go down. If it is throw n up in the air, it is know
that
it
will
travel
upwards
over
a
certain
time
interval;
but
afterwards will come down.
2. Random Experiment
With random experiments, the results cannot be predicted,
since they depend on chance.
Examples
If a coin is flipped, it is not known beforehand whether it will
be heads or tails.
Similarly, if a die is rolled, the result cannot be determined
beforehand.
3. Probability Theory
Probability theory
deals with assigning a number to each
possible result that can occur in a random experiment. With this
being said, the following definitions need to be introduced:
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PROBABILITY Theory (I)
IES Alfonso X Murcia Matemáticas 4º ESO
Outcome
A
outcome
is
each
of
the
possible
results
of
a
random
experiment.
Obtaining heads when flipping a coin.
Obtaining a 4 when rolling a d ice.
Sample Space
The
sample
space
is
the
set
of
all
possible
outcomes
of a
random experiment. It is denoted by S (or by the Greek letter Ω).
Sample space of a coin:
S = {H, T}.
Sample space of a die:
S = {1, 2, 3, 4, 5, 6}.
Event
An event is any subset of the sample space.
For
example,
when
rolling
a
die,
an
event
would
be
the
outcome of an even number, or another, obtaining a multiple of 3.
Example
A
bag
contains
blue
and
red
balls.
Three
balls
are
drawn
successively. Calculate:
1. The sample space.
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IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
S = {(b,b,b); (b,b,r); (b,r,b); (r,b,b ); (b,r,r); (r,b,r); (r,r ,b);
(r, r,r)}
2. The event A = (draw three balls of the same color).
A = {(b,b,b); (r, r,r)}
3. The event B = (extract at least one blue ball).
B= {(b,b,b); (b,b,r); (b,r,b); (r,b,b); (b,r,r); (r,b,r); (r,r ,b)}
4. The event C = {extract only one red}.
C = {(b,b,r); (b,r,b); (r,b,b)}
Events
Elementary Event
An elementary event is one of the elements that make up the
sample space.
For example, if a die is thrown, an elementary event would be
a 5.
Compound Event
A compound event is any subset of the sample space.
For example, if a die is thrown, a compund event would be an
even number, another, a multiple of 3.
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IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
Sure Event
The sure event, S, is formed by all possible results (that is to
say, the sample space).
For example, rolling a die and obtaining a score of less than 7.
Impossible Event
The impossible event,
, does not have an element.
For example, rolling a die and obtaining a score of 7.
Disjoint Events or Mutually Exclusive
Two events, A and B, are mutually exclusive when they don´t
have an element in common.
If outcome A is to obtain an even number from a die and B is
to obtain a multiple of 5, A and B are mutually exclusive events.
Independent Events
Two events, A and B are independent if the probability of the
succeeding event is not affected by the outcome of the preceeding
event.
By rolling a die twice, the results are independent.
Dependent Events
Two events, A and B are dependent if the probability of the
succeeding
event
is
affected
by
the
outcome
of
the
preceeding
event.
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IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
For example, two dependent events would be drawing two cards
from a deck (one at a time) without redepositing them.
Complementary Event
The complementary event of A is another event that is realized
when A is not realized. It is denoted by
or A'.
For example, the complementary event of obtaining an even
number when rolling a dice is obtaining an odd number.
4. Probability Properties
Probability Axioms
1.The probability is positive and less than or equal to 1.
0 ≤ p(A) ≤ 1
2. The probability of the sure e vent is 1.
p(S) = 1
3.If A and B are mutually exclusive, then:
p(A
B) = p(A) + p(B)
Probability Properties
1
The
sum
of
the
probabilities
of
an
event
and
its
complementary is 1, so the probability of the complementary event
is:
2 The probability of an i mpossible event is zero.
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PROBABILITY Theory (I)
IES Alfonso X Murcia Matemáticas 4º ESO
3 The probability of the union of two events is the sum of their
probabilities minus the probability of their intersection.
4 If an event is a subset of another event, its probability is
less than or equal to it.
5 If A1, A2, ..., Ak are mutually exclusive between them, then:
6 If the sample space S is finite and an event is S = {x 1, x2,
..., xn} then:
For
example,
the
probability
of
obtaining
an
even
number,
when rolling a die, is:
P(even) = P(2) + P(4) + P(6)
Pr o b a b i l i t y Fo rm ul a
If
a
random
elementary
experiment
events,
all
is
equally
conducted
likely,
and
in
A
which
is
an
there
are
event,
n
the
probability of event A is:
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IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
Examples
Find the probability of tossing two coins simultaneously and
obtaining two heads.
Possible outcomes: {hh, ht, th, tt}.
Favorable outcomes: 1.
In
a
deck
of
40
cards,
if
one
card
is
removed,
find
the
probability of an Ace ( P(Ace)) being drawn and the probability of a
Diamond (P (Diamond)) being drawn.
Possible outcomes: 40.
Favorable outcomes of aces: 4.
Favorable outcomes of diamonds: 10.
Calculate the probability of rolling a die and obtaining:
1 An even number.
Possible outcomes: {1, 2, 3, 4, 5, 6}.
Favorable outcomes: {2, 4, 6}.
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IES Alfonso X Murcia Matemáticas 4º ESO
PROBABILITY Theory (I)
2 A multiple of three.
Favorable outcomes: {3, 6}.
3 A result greater than 4.
Favorable outcomes: {5, 6}.
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