Math 9th grade
LEARNING OBJECT
LEARNING UNIT
Resolution of random situations, using Laplace’s
Collect and analyze data, rule of succession.
before coming to your own
conclusions.
S/K
SCO: Calculate odds on equiprobable spaces.
 Identify
random
equiprobable
experiments described in various sources.
 Make a count of all the possible events in
a random equiprobable experiment.
 Make a count of all the possibilities of a
specific
event
occurring
in
a
random
experiment.
 Enter the probability as a ratio of the
chances of an event and the total number of
possible events.
 Express
percentages.
Language
Socio cultural context of
the LO
Curricular axis
Standard competencies
Background Knowledge
Basic Learning
Rights
English Review topic
Vocabulary Box
probabilities
in
terms
of
English
Classroom, family meetings, situations related to
gambling.
Random thinking and data systems
Use basic concepts of probability (sample space,
event, independence, etc.).
Probability, sample space, random experiments, and
events.
Recognize the notions of sample spaces and events,
like the notation P (A) for the probability of an event
occurring.
 Occurrence: Something that happens (noun)
 Chance: An occasion that allows something to
English Review Topic
occur (noun)
 Dice: A small cube with a different number of
spots on each side, used in games involving
chance (noun)
 Toss: to throw something carelessly (verb)
 Bingo: a game in which prizes can be won by
matching numbers on a card with those chosen
by chance (noun)
Conditionals
NAME: _________________________________________________
GRADE: ________________________________________________
Introduction
Mathematical Probability and Gambling
Every person has, at one point in their life, been associated either
directly or indirectly with a game of chance, whether it be table games
in casinos, at family gatherings, or other events. It is chance itself
which generates entertainment and fun for playing.
Games with dice, for example, in which each player has the same
opportunities to either win or lose, are equiprobable events, which
indicates that they can be studied and even predicted, given enough
time.
Calculating probability leads us to think of a simple formula, called
Laplace’s rule of succession, named for the French mathematician,
astronomer, and physicist Pierre Simon Laplace.
However, not all forms of gambling represent equiprobable events. We
must first establish the conditions of play and realize the nature of the
events that may occur in the game to determine if the game is
equiprobable or not.
Objectives
 To solve probability situations using Laplace’s rule, as well as
counting.
 To calculate probabilities of event spaces with equal probability.
Activity 1
SCO: Calculate odds on equiprobable spaces.
Skill 1. Identify random equiprobable experiments described in
various sources.
Skill 2. Make a count of all the possible events in a random
equiprobable experiment.
Skill 3. Make a count of all the possibilities of a specific event
occurring in a random experiment.
What does it
equiprobable?
mean
that
a
probabilistic
gambling
situation
is
All gambling can be studied by probability, but not all games have the
feature of containing an equiprobable sample space.
To solve the question, we must recall certain basic concepts, such as
probability, random experiments, sample spaces, and events.
Random Equiprobable Experiments
We'll start by saying that probability is the calculation of the various
possibilities that occur before a random experiment. For example:
The football team Athletic Palms has played ten matches and only lost
one. We could calculate that in their next game there is a high
probability of them winning, since in their last ten games they only lost
once.
In probabilistic study we speak of randomized experiments, these
being events or situations in which you cannot predict the results with
certainty. This depends on the circumstances of chance.
A sample space is what we call all the possible outcomes expected
from a random experiment. Depending on the circumstances, there
may be multiple options. All options are then represented as a whole
set.
However, in the sample space which we consider as a whole, certain
subsets are also determined, which are called events. Thus, an event
is contained within a set of options which are emerging from a sample
space and are represented as one or more particular situations
There are various types of events: simple events, compound events,
secure events, and impossible events.
In short, we can say that a random equiprobable experiment happens
when all the elements of the sample space have an equal chance of
being chosen and, therefore, have the same probability of occurrence.
Contemplate
Is it possible to calculate the probability of the throwing of a die
resulting in an even number?
Let us analyze the following examples:
1. Verify in each situation whether or not it corresponds to a
equiprobable event.
a. In the parties of the neighborhood 15 de abril, a raffle to win a
refrigerator is held annually, which consists of drawing a red
ballot from a ballot box, as shown in Figure.
The preceding figure shows the different possibilities of a person
reaching into the ballot box and choose a ballot that is red, green, blue
or purple.
Let us analyze each possibility:
● Possibility of choosing a red ball=
𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
● Possibility of choosing a green ball=
● Possibility of choosing a blue ball=
1
=9
𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
● Possibility of choosing a purple ball=
3
=9
2
=9
𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑐𝑎𝑠𝑒𝑠
3
=9
In conclusion, just as choosing any ball does not generate the same
possibility of choosing that ball again, this situation does not
correspond to an equiprobable event.
a. Before starting a professional soccer game, the referee carries
out a coin toss with the captains of each team to decide which
team starts with possession of the ball.
As we see in this situation, a coin toss has two possible conclusions: to
fall either heads-up or tails-up. When a player chooses face, he/she
has the same chance of winning than the one who chooses tails. Thus,
we can say that this situation is a random, equiprobable experiment.
1. The tossing of a six-sided die on a board game is carried out.
Determine the answers to the following situations:
a. Does the tossing of a six-sided die represent an equiprobable
event? If so, determine the sample space.
b. Keeping in mind the sample space, what possible events could
occur in the coin toss? Indicate what type of event this would
represent.
Solution
a. The launch of a 6-sided die is an equiprobable experiment,
because each of the six numbers between 1 and 6 has the same
chance of coming out face-up. Now, we determine the sample
space, which is symbolized by the letter S.
As we talk about the toss of a six-sided die, the possible results
would be the following:
S={1, 2, 3, 4, 5, 6}
b. The events are symbolized with capital letters and are classified
as either simple, compound, impossible, or secure events. Now, let's
look at the following events that could occur after throwing a die:
● The throwing of a die resulting in a prime number:
A= {2, 3, 5} This would represent a compound event, as it has more
than one element.
● The throwing of a die resulting in a number with four divisors:
B= {6} This would represent a simple event, as it only has one
element.
● The throwing of a die resulting in a number greater than six:
C = { } This would represent an impossible event, as it has no
elements.
● The throwing of a die resulting in a natural number:
E= {1, 2, 3, 4, 5, 6} This would represent a secure event, as it
contains all of the elements of the sample space.
Collaborative Learning Activity
Based on the preceding information, carry out the following activity in
groups
of
three
students:
1. Please indicate which of the following random experiments are
equiprobable:
a. To choose at random a two-digit number of a lottery, ranging
from 00 to 99.
b. To guess the four digit code of a credit card.
c. Choosing a motorcycle at random from a dealership.
d. The tossing of two dice along with the tossing of a coin.
e. To choose at random a ball from an urn which has five green
balls, two red balls, and one black ball.
2. In a playground game, each player always throws two six-sided
dice each.
Given the above information, solve the following:
a. Why is the throwing of two dice a random equiprobable
experiment?
b. Find the sample space of the throwing of two dice.
c. Determine and classify the following events occurring at the
throwing of two dice.
Please indicate:
● The event that the sum of their faces is an even number.
● The event that the sum of their faces has two divisors.
● The event that the sum of their faces is an odd number.
● The event that the sum of their faces is seven.
● The event that the sum of their faces is greater than or equal to
the number nine.
● The event that the sum of their faces is less than six.
● The event that one face shows a six.
● The event that both faces show the same number.
Activity 2.
Skill 6. Enter probability as a ratio of the chances of an event and the
total number of possible events.
Skill 7. Express probabilities in terms of percentages.
What is the probability that the spinner stops on an even
number?
By rotating the wheel we are sure you will stop at one of the 8
numbers. However, how would we bet that the roulette stops on a
number of our choosing? This is why we study probability: to get an
idea of the possible outcomes that could happen in a random event.
Probability is then a statistical tool that allows us to approximate what
might happen in a random event, considering that they are under
normal conditions.
Probability
According to the wheel in the previous image, let's answer the
following question: What is the probability that the spinner will stop on
an even number?
Before answering the question, remember the following: In an
equiprobable random experiment, we can calculate the probability of
an event using the ratio between the number of elements in the event
and the number of elements in the sample space of a random
experiment study. This is known as Laplace's rule.
Now, considering roulette, which is a equiprobable experiment, the
sample space would be:
S= {1, 2, 3, 4, 5, 6, 7, 8}
And in the event study of the probability that it stops on an even
number, the elements would be:
A= {2, 4, 6, 8}
Taking into account the mathematical definition of probability:
#𝐴 4
𝑃 = #𝑆 =8
It indicates that in eight tosses, there is a probability that it stops on
an even number four times.
Now if we make the ratio between four and eight, we have:
#𝐴 4
𝑃 = #𝑆 =8 = 0.5
Multiplying 0.5 times 100, the product is 50, which tells us that 50% of
the time the roulette wheel will stop on an even number.
Consider another example, where two randomized experiments are
simultaneously involved. If we drop an eight-face die and a coin, what
would be the sample space? What would be the probability that tossing
the die and the coin at the same time would result in a multiple of
four?
Given that the sample space of the eight-sided die would be:
S={1,2,3,4,5,6,7,8}
And that the sample space of the coin would be:
S={heads, tails}
Now, if you were to toss the coin and the die simultaneously, the
sample space that it generated would be as follows:
S={(1,c), (1,s), (2,c), (2,s), (3,c), (3,s) (4,c), (4,s) (5,c), (5,s), (6,c),
(6,s), (7,c),(7,s) (8,c),(8,s)}
Now, let us find the elements of the event in which the tosses result in
a multiple of four:
B={(4,s),(4,c), (8,s),(8,c)}
Applying the mathematical definition of probability:
𝑃=
#𝐵
#𝑆
4
=16
This shows us that there is a probability that stops 4 times on a
multiple of 4 in 16 tosses.
Now if we make the ratio between 4 and 16, we have:
𝑃=
#𝐵
#𝑆
4
=16 = 0.25
Multiplying 0.25 times 100, the result is 25, which tells us that there is
a 25% chance of getting a multiple of four.
Learning Activity
1. In a board game two dice of 6 faces are tossed continuously. The
player who throws and gets a pair, or gets seven as the sum of
both dice, gets to throw again. What then is the probability of
getting seven doubles?
2. A professor of mathematics, to teach the class probability,
carries a deck of 52 poker cards, which are divided into 4
groups, according to their suits. Each group of thirteen cards are
numbered from 2 to 10 and also include four face cards, with
letters A, J, Q, and K, as shown in the figure.
According to the preceding context:
a. Find the probability of getting a four of diamonds.
b. Find the probability of getting an odd-numbered club.
c. Calculate the probability of drawing a pair of any card.
d. Determine the probability of drawing a jack of any suit.
Abstract
In our daily lives we are constantly faced with events in which we are
not certain of what can happen. When we buy a ticket from a raffle of
two numbers, for example, we know it is difficult to win the prize.
However, mathematically we know that there are a hundred options.
This common equiprobable random experiment shows us how we
interact on a daily basis without realizing the concept of probability.
Randomized equiprobable experiments, as we know, are those in
which any element of the sample space is equally likely to result, such
as with playing cards, dice, and bingo, among others. The sample
space is determined according to the contextualized situation and is
defined by all possible options which can occur in a random
experiment. For example, if we throw a die, we know that there are six
possible outcomes
Now, knowing the sample space of a random experiment, we can
establish from them certain events, which would then be subsets of
the sample space. For example, with the toss of the dice, we can
determine the probability of getting a prime number, a pair, or an odd
number, among other results.
When we have defined, in a random equiprobable experiment, the
sample space and the event that we want to happen, we can then talk
about probability. The probability is then defined as the ratio of the
number of possible outcomes of an event and the number of elements
in the sample space. This mathematical procedure is known as the
Laplace's rule.
Finally, the random equiprobable experiments can be calculated and,
at times, we can find the certainty of the occurrence of a specific
event.
Homework
A professor of statistics performed a randomized experiment in the
classroom, tossing a die and a coin at the same time and recording the
results.
According to the above context:
a) Find the sample space that determines the situation.
b) Find the probability of getting a pair and a heads-up.
c) Calculate the probability of getting a prime number and a tailsup.
d) Find the probability of getting a natural number and a heads-up.
e) Find the probability of getting a number with three divisors.
Evaluation
Answer questions 1, 2, and 3, according to the following context:
In a board game a 12-sided die is used, as shown below:
1. The event in which throwing the dice results in a prime number,
is determined by the set:
a.
b.
c.
d.
A={1 , 2, 3, 5, 7, 9, 11}
B={1, 2, 3, 4, 7, 9, 11}
C={2,3,5,7,11}
D={1,2,4,7,9,12}
2. The probability that a dice roll results in a number divisible by
three is approximately:
a.
b.
c.
d.
30%
40%
33.3%
18.7%
3. The probability that a dice roll results in an integer is:
a.
b.
c.
d.
50%
70%
90%
100%
4. Complete the sentence
Laplace's rule states that the probability of an event occurring is given
by the ratio between the _____________________and
______________________.
5. Indicate whether the following statements are true (T) or false
(F).
a. The sample space is the set of all the possible outcomes
that could be generated in a random experiment.
b. Laplace’s rule can be utilized to calculate the outcome of
any random experiment.
Bibliography
● De Armas J., Ramírez M., Acosta M., Romero J., Gamboa J.,
Rojas V., Chappe., Morales D., Salazar F,. (2.014). Matemáticas
9. Colombia: Editorial Santillana S.A.
● Espacio muestral equiprobable. Fecha de consulta mayo 20 de
2016.
Retrieved
from:
http://www.ceibal.edu.uy/userfiles/P0001/ObjetoAprendizaje/HT
ML/Sucesos%20equiprobables_SRealini.elp/espacio_muestral_eq
uiprobable.html
● Ortiz L., Ramírez M., Joya A., Celi V., Acosta M., Perdomo A.,
Morales D., Gamboa J., (2013).
Matemáticas 7. Colombia:
Editorial Santillana S.A.
English Review Topic: Conditionals
Complete the following sentences by choosing the correct form of the
verbs shown in parenthesis.
1. If I
(to study), I
(to
pass) the exams.
2. If the sun
we
3. If he
he
4. If my friends
I
5. If she
she
6. If we
we
7. If you
you
8. If Rita
teacher
(to shine),
(to walk) to the town.
(to have) a temperature,
(to see) the doctor.
(to come),
(to be) very happy.
(to earn) a lot of money,
(to fly) to New York.
(to travel) to London,
(to visit) the museums.
(to wear) sandals in the mountains,
(to slip) on the rocks.
(to forget) her homework, the
(to give) her a low mark.
9. If they
(to go) to the disco,
they
(to listen) to loud music.
10. If you
(to wait) a minute,
I
(to ask) my parents.