Math 9th grade
LEARNING OBJECT
LEARNING UNIT
I collect and analyze data,
and obtain my own
conclusions
S/K
Identification of the properties of probability
 Recognizes number one as the probability of the sample
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Language
Socio cultural context of
the LO
Curricular axis
Standard competencies
Background Knowledge
Basic Learning
Rights
English Review topic
Vocabulary box
space.
Recognizes zero as the probability of an impossible
event.
Identifies and explains that the sum of the probabilities
of the events of the sample space equals one.
Deduces that the probability of an event is a number
between zero and one.
Recognizes that the union of events is an event.
Recognizes that the probability of the union of two events
is the addition of the probabilities of each event, and
writes down made up examples.
English
Classroom and academic institution
Random thought and data systems.
I speculate on the result of a random experiment,
using proportionality and basic concepts of probability.
Fraction as reason and percentage, operations
between sets, probabilistic concepts of population,
sample and event.
Recognizes the notion of sample space of events, and
the notation P(A) for the probability of event A.
A review of demonstrative pronouns
 Outcome (noun): a result or effect of an
action, situation, etc.
 Attempt (noun): the act of trying to do
something, especially something difficult.
 Set (noun): In mathematics, a set is a group of
objects with stated characteristics.
 Carry out (verb): to do or complete
something, especially that you have said you
would do or that you have been told to do.



Suit (noun) (playing cards): any of the four
types of card in a set of playing cards, each
having a different shape printed on it.
Ace (noun) (playing card): one of the four
playing cards with a single mark or spot. The
ace has the highest or lowest value in many card
games.
Measurement (noun): a value, discovered by
measuring, that corresponds to the size, shape,
quality, etc. of something.
Reference:
http://dictionary.cambridge.org/
NAME: _________________________________________________
GRADE: ________________________________________________
Introduction
The teacher presents an animation that shows the concepts that make part of the
solution of probability situations, like: experiment, which refers to all action that is
going to be measured; sample space, which indicates the set of all possible results
that may occur when carrying out an experiment; and events, which refer to any
subset of the sample space that contains the favorable results of the experiment.
There are different types of events that can appear when solving exercises related
to probability, but the three that are most common are:
Probable event: made up from results that are possible to obtain, that is, from the
elements that make part of the sample space.
Impossible event: events that have no elements, which means that they don’t make
part of the sample space.
Sure event: formed by all the possible outcomes, that is, by the sample space.
Example:
A change purse has 2 coins of $50, 4 of $100 and 6 of $200.
The probability of taking out a coin with a value that is a multiple of 10 is:
P(x) =
cases of success
Sample space
𝑃(𝑥) =
12
=1
12
The probability of taking out a $500 coin is:

cases of success
P(x) =
Sample space
𝑃(𝑥) =
0
=0
12
The probability of taking out a $200 coin is:
𝑃(𝑥) =
𝑐𝑎𝑠𝑒𝑠𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑆𝑎𝑚𝑝𝑙𝑒𝑠𝑝𝑎𝑐𝑒
𝑃(𝑥) =
6
= 0.5
12
Objectives
Recognizing the properties that probabilities satisfy
Activity one
Skill
Recognizes number one as the probability of the sample space.
Recognizes zero as the probability of an impossible event.
EXTREME VALUES OF EVENTS PROBABILITY
When talking about probability, one must keep in mind that events can be: sure,
possible (probable) or impossible (improbable). To make the calculations of
probability easier, the following properties can be applied:
𝑃(∅) = 0. So that if an event that isn’t in the sample space is considered, the
probability of it occurring is 0.
𝑃(𝐴) = 1. If all the sample space is considered, the probability that something
in the sample space occurs is 1. (Flórez, I. & Ramírez, C. 2011, p.123)
Example:
10 students are selected in a classroom, and are asked for their last names. These
are written down in pieces of paper that are put in a box. Taking into account that
the last names of the selected students are:
Álvarez
Parra
Castro
Pérez
Cortes
Rodríguez
Díaz
Rodríguez
Ortiz
Zapata
Sure event:
The probability of taking out a paper from the box with a last name that starts with A,
C, D, O, P, R or Z is:
𝑃(𝑥) =
𝑐𝑎𝑠𝑒𝑠𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑆𝑎𝑚𝑝𝑙𝑒𝑠𝑝𝑎𝑐𝑒
𝑃(𝑥) =
10
=1
10
Probable or possible event:
The probability of picking a paper from the box that has the last name Rodríguez on
it is:
𝑃(𝑥) =
𝑐𝑎𝑠𝑒𝑠𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑆𝑎𝑚𝑝𝑙𝑒𝑠𝑝𝑎𝑐𝑒
𝑃(𝑥) =
2
1
= = 0.2
10 5
Impossible event:
The probability of taking out a paper from the box that has the last name Sánchez
written on it is:
𝑃(𝑥) =
𝑐𝑎𝑠𝑒𝑠𝑜𝑓𝑠𝑢𝑐𝑐𝑒𝑠𝑠
𝑆𝑎𝑚𝑝𝑙𝑒𝑠𝑝𝑎𝑐𝑒
𝑃(𝑥) =
0
=0
10
Learning activity
Complete the sentence:
1. Action that is measured. ______________________.
2. Set of possible results. ________________________.
3. Event that has as outcomes the same number of elements as the sample
space.__________________
4. Event that has as
space._________________
an
outcome
some
element
of
the
sample
5. Event that has as outcomes elements that do not belong to the sample
space.______________________
6.
Subset
of
the
sample
results.__________________
space
that
contains
the
favorable
Learning Activity:
Relate each case to the type of event that corresponds to it, based on the following
situation:
A survey is made to the 9th grade students, to ask them what type of pet they would
like to have. The results are the following:
Table 1
Pet
Dog
Cat
Tortoise
Hamster
# of students
12
8
3
5
Table 2
a. The probability that students want to
have a four-legged pet.
( )Probable event
b. The probability that students want to
have a dog.
( )Impossible event
c. The probability that students want to
have a fish.
( )Sure event
Activity Two
Skill
1. Identifies and explains that the sum of the probabilities of the events of the
sample space equals one.
2. Deduces that the probability of an event is a number between zero and one.
ADDITION OF PROBABILITIES
When the addition of events is equal to the sample space, these events are
considered complementary. So:
𝑃(𝐴) + 𝑃(𝐵) = 𝛺
Where A and B are events that can happen in sample space 𝛺.
Example
You have, in a box, 20 ping pong balls of three colors: 8 red, 7 green and 5 blue.
What is the probability that you will draw a red one on the first attempt? And what is
the probability of drawing a green on? Or a blue one?
𝐵𝑎𝑙𝑙𝑠𝑜𝑓𝑡ℎ𝑒𝑐ℎ𝑜𝑠𝑒𝑛𝑇𝑜𝑡𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑏𝑎𝑙𝑙𝑠 ∈ 𝑡ℎ𝑒𝑏𝑜𝑥
𝑃(𝑥) =
So, the probability of picking a red ping pong ball is:
8
= 0.4
20
𝑃(𝑅) =
A green one:
𝑃(𝑉) =
7
= 0.35
20
𝑃(𝑅) =
5
= 0.25
20
And a blue one:
And, when you add the probabilities obtained for each color, you obtain the sample
space, as follows:
𝑃(𝑅) + 𝑃(𝑉) + 𝑃(𝐴) =
8
7
5
20
+
+
=
=1
20 20 20 20
Or, equally:
𝑃(𝑅) + 𝑃(𝑉) + 𝑃(𝐴) = 0.4 + 0.35 + 0.25 = 1
Let’s keep in mind that:
The results obtained in each probability are fractions that can be expressed as
decimals that will have a value from zero to one. Hence, when you add the
probabilities as decimals, you will also obtain the sample space.
Learning activity
Match the complementary events, that is, the events for which, when you add their
proportions, the result is the sample space.
a. The probability that, when
rolling a die, the result will be an
even number.
( ) The probability that, when rolling a
die, the result won’t be a multiple of 3.
b. The probability that, when
rolling two dice, the result will be
greater than 8.
( ) The probability that, when rolling two
dice, the result will be smaller than 8.
c. The probability that, when rolling
a die, the result will be a multiple
of 3.
( ) The probability that, when rolling a
die, the result will be an odd number.
Now, explain (in the notebook, and using demonstrative pronouns) what events are,
what sample space is, and why the addition of the probabilities of the events equals
the sample space (give an example).
Very important: The closer the probability value is to zero (0), the smaller the
possibility that the event will happen; and, the closer the probability value is to one
(1), the greater the possibility that the event will happen.
Learning activity
Answer if each of the following situations is true or false:
1. When you pick a card from a deck of poker cards (made up by 52 cards, divided
into 4 suits), the probability of getting a card of spades is 𝑃(𝑝) = 0.25 _______
2. In a box there are 10 blue balls, 5 red balls and 8 yellow balls. The probability of
picking a yellow ball is 𝑃(𝐴) = 0.043 _______
3. When you select a card from a deck of poker cards (consisting of 52 cards,
divided into 4 suits), the probability of getting an ace is 𝑃(𝐴) = 0.019 _______
4. When you roll a die of six faces, the probability of obtaining a number smaller
than 7 is 𝑃(𝐷) = 1 _______
5. There are 7 blue, 8 red and 5 yellow balls in a box. The probability of picking a
yellow, red or blue ball is 𝑃(𝑋) = 1 _______
Activity three
Skill
3. Recognizes that the union of events is an event.
4. Recognizes that the probability of the union of two events is the addition of the
probabilities of each event and writes down made up examples.
UNIÓN OF PROBABILITY EVENTS
The probability of the union of two events is seen as the addition of the probabilities
of each of the events. The way in which the union is made depends on whether the
events are compatible or incompatible.
The way in which the probabilistic union can be made, according to the case, is
presented next:
 Union of incompatible events:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
 Union of compatible events:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Examples:
Learning activity
Complete the following sentences:
Two events are ______________ if they have any element in common.
Two events are _____________ if they have no elements in common.
Learning activity
The experiment of picking a card from a deck of poker cards that has 52 cards
divided into 4 suits is carried out, and the following events are considered:
A= Getting an ace
B= Getting a J
C= Getting a letter (J, Q, K, A)
D= Getting a number smaller than 6
E= Getting an even number
F= Getting an odd number
Answer true or false:
1. The probability 𝑃(𝐴 ∪ 𝐵) is compatible.
2. The probability 𝑃(𝐵 ∪ 𝐷) is approximately 0.38.
3. The probability 𝑃(𝐷 ∪ 𝐸) is compatible.
4. The probability 𝑃(𝐹 ∪ 𝐴) is incompatible.
9
5. The probability 𝑃(𝐸 ∪ 𝐹) is 52 .
6. Match the left and right columns:
( )
2
13
( )
4
13
The probability 𝑃(𝐶 ∪ 𝐷) is equal to
( )
0.38
The probability 𝑃(𝐵 ∪ 𝐶) is equal to
( )
0.61
The probability 𝑃(𝐴 ∪ 𝐷) is equal to
The probability 𝑃(𝐴 ∪ 𝐵) is equal to
7. By carrying out an experiment with two dice, you can get different results. Suggest
two problem situations for the following events.
Compatible event: ____________________________________
____________________________________________________.
Incompatible event: ___________________________________
______________________________________________________
Abstract
Probability is a statistical study that is very important since it allows us to know the
occurrence of events in different experiments that are usually carried out.
When talking about probability, we make reference to a number between zero and
one, which can be represented as a fraction or a decimal, and not of the possibility
that an event happens, which indicates whether or not it can occur.
All events can be in one of the following three categories:
● Probable: makes reference to all the events that are in the sample space.
● Impossible: are events that do not make part of the sample space.
● Sure: corresponds to all of the sample space.
With statistical probabilities, operations that meet the own properties of sets can be
carried out. So, when the probabilities of all the events of an experiment are added
up, the result is one, which is equivalent to the sample space.
Also, the union of probabilistic events can be carried out. This property is divided
into two types:
● Union of compatible events: corresponds to the adding of the probabilities of the
events, minus their intersection. These unions are solved with this formula:𝑃(𝐴 ∪
𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
● Union of incompatible events: the adding of events that don’t have elements in
common, by the addition of their probabilities:
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
This union allows to know the probability of occurrence of several events that can
appear in one same sample space, and it is represented by the sum of the
probabilities of these events, which will always be between zero and one. In other
words, the union of several events corresponds to an event.
Homework
In groups, students must solve the following problem situations:
1. Indicate which type of event corresponds to each one of the following cases:
(probable, improbable, sure)
a. In a box there are 4 red balls, 5 green and 6 purple. The probability of taking
5 red balls out is…
b. When rolling two dice, the probability of getting a result smaller than 13 is…
c. When picking a card from a deck of poker cards, the probability of it being an
ace is…
2. Solve the following problem situations with incompatible events:
a. Of the 15 men in the room, 10 have brown eyes, and of the 12 women, 8 have
brown eyes. What is the probability of selecting 5 men or 4 women with brown
eyes?
b. In a box, there are cards numbered from 10 to 25 (that is, 10, 11, 12…25).
What is the probability, if you pick a card at random, that the sum of its digits
is 3 or 4?
c. If you roll a pair of common dice, what is the probability of obtaining a
minimum of 10 in the sum of the numbers given by each die?
3. Solve the following problems with compatible events:
a. A whole number between 1 and 9 is picked. What is the probability that the
number is a multiple of 2 or 5?
b. A die of 6 faces is rolled. What is the probability that it will show an odd
number or a number smaller than 5?
c. A ball is picked at random from a raffle drum with balls numbered from 1 to
36. What is the probability that the ball has an odd number or a number
smaller than 12?
Bibliography
● Fernández, S., Cordero, J. y Córdoba, A. (2002). Estadística Descriptiva (2a. ed.)
Madrid: ESIC.
● Flores, I. y Ramírez, C. (2011).Didáctica de la estadística (1a. ed.) Bogotá D.C.:
Ediciones USTA.
● Freund, J, y Simon, G. (1992). Estadística Elemental (8a. ed.) México: Pearson
Educación.
● Sampieri, R., Fernández, C. y Baptista, M. (2014). Metodología de la
Investigación (6a. ed.) México: Mc Graw Hill.
● www.virtual.unal.edu.co (s.f.) Distribución de probabilidad. Recuperado el 8 de
marzo
de
2016,
de
http://www.virtual.unal.edu.co/cursos/ciencias/2001065/html/un2/cont_215_57.
html
● www.proyest1.blogspot.com.co (s.f.) Probabilidad y estadística.
Recuperado
el
8
de
marzo
de
2016,
de
http://proyest1.blogspot.com.co/p/blog-page_3746.html