Peluang (Part 1) | Definisi Peluang, Komplemen Kejadian dan Frekuensi Harapan Matematika Kelas 12

Peace be upon you, and Allah's mercy and

blessings. Meet me again,

Deni Handayani, on the Matlab channel. This

is a video discussing the

first part of the opportunity material. In this first part of the video

we will learn the definition of the

probability of the complement of an event and the

expected frequency.

God willing, I will discuss the material on compound events in the

second part of the video. Okay,

let's just discuss the material.

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Okay, now let's discuss the

first part of the probability material. We start from the definition of

opportunity. Opportunity is the likelihood of

an event occurring.

Well, mathematically we can calculate this probability

. The formula for the probability of an

event A occurring is like this. P =

NA/ NS. Where PA is the probability of

event A. Na is the number of ways

or the number of possibilities for A to occur and

NS is the number of all possibilities.

We can find NA and NS using

the concept of enumeration rules that we

learned in the previous video.

Friends, you still remember about

permutations and combinations, we will

use them again in this opportunity material. Well, the

range of this opportunity is between 0

and 1. The probability of an event is

greater than or equal to 0 and

less than or equal to 1. When

the probability of an event is 0, it means that the

event is impossible to happen and when

the probability is one, it will definitely happen.

To be clearer, friends,

pay attention to the following example. A

die is thrown once. What is

the probability that the sum of the dice

is a prime number? Okay,

guys, you know that dice have six

sides, right? So, if we throw a

dice, the possible numbers that appear are

like this. 1 2 3 to 6. Yes,

here what is being asked is the probability of the

number of dice appearing being a

prime number.

For example, event A is the event that

the number of dice faces appears to be a

prime number. Let's see which are the prime numbers from 1 to 6

. This is 2

then 3 is prime too and 5 is prime

too. So how many possibilities are

there that A is a prime event?

There are three, right? The numbers 2, 3 or 5. So

Na is the number of possibilities A is the number of

possibilities for a prime number to appear

is 3. Now we determine NS. NS

is the number of all possibilities when

we throw a dice, the number of

all possibilities is 6, right? The

numbers 1, 2, 3, or 6 can appear, right? So

the chance of a prime number appearing means NA/NS. PA

= NA / NS. How much is 3/6? 1/2.

So, the probability of the number of dice appearing

being a prime number

is 1/2.

Okay, let's move on to the second example to make it

clearer. Okay, now let's discuss the

second example. From those five men and

five women, four people will be selected

at random. What is the probability that of the four

people selected, three are

men and one is a woman? Okay,

now let's try to solve

this problem. There are seven

men and five women here. So, I'll just write it

like this, 7P 5W.

Meanwhile, 4 people were chosen, right?

We will choose four people. And

the question is what are the chances of

these four people? Three of them are

men. So, out of these four people, what is

the probability that three

men and one woman will be chosen? OK, let's say

event A is the event of selecting three

men and one woman. We will look for

Na. So the number of ways chosen

is three men and one woman. So

here we use a combination,

guys. Why combination? We will

choose three men from the seven men

available.

When we choose 3 out of 7, we

don't pay attention to the order, right? So

we use a combination of 7 taken by 3.

Then we will choose one woman

from the five women that exist times the combination

of 5 taken by 1. Okay. So, then

to determine the ns. To

determine ns we will choose 4 from

how many? Of all these people in total. 5 +

7 is 12 right? So we will choose four

people from the 12 available without

paying attention to whether they are male or

female. So the NS is many, all

the possibilities are to choose 4 out of the 12

people there are. Okay, now we will

calculate the odds. The probability of event

A or the probability of selecting three men and

one woman is Na / NS. This NA is a

combination of 7 taken 3 times, the

combination of 5 taken 1. Then

this NS is a combination of 12

taken 4 times. Now we use

the combination that we have learned.

The combination of n taken r, still remember

? N factorial/ N - R factorial * R

factorial. This is the combination formula. So

the combination of 7 taken 3 = 7

factorial. 7 - 3 4 we factorize

3 factorial. The combination of 5 taken 1 is

5 factorial divided by 5 - 1 is 4

factorial * 1 factorial then divided by

the combination of 12 taken by 4. 12

factorial eh per 12 - 4 is 8 factorial

* 4 factorial. Okay. Well, now we

change the 7 factorial, we change it to multiply it

backwards, okay? 7 * 6 * 5 * 4 factorial.

Stop here because there is already the

same one below. So we'll just cross this out.

Then 5 factorial is also the same. We

change this to 5 * 4 factorial.

Let's just cross this out too. Oh yes, 3 factorial

means 3 * 2 * 1, what is

the value? It turns out this is 6. So we

also cross this out with 6.

Then we change the bottom 12 factorial

to 12 * 11 * 10 * 9 * 8

factorial. Stop here because there is

already the same one below. We cross out 4 *

3, which is 12. We cross out 12.

Then we divide 10 by 2, which becomes 5

. Well, now we get 7 * 5 this is

35 this is 5 yes. Then the bottom one is 11 *

5 * 9. So 35 * 5/11 * 5 * 9. This is the

same as 5, we cross it out, it is also the same as

35/11 * 9, which is 99.

So the answer is the probability is

35/99.

For more challenging questions,

please see the link in the description.

I will discuss five questions about the

opportunities that arise in university selection

such as UTBK, SBMPTN or

SIMAK UI. Okay, now we move on to the

next sub-topic, namely the complement of

an event.

If the probability of an event A is PA,

then the probability of the complement of event A is

PA^C or PA AK'. This is

complementary writing. The meaning of the complement of an

event is the opposite event,

friends. For example, when we throw

a dice, if A is the event of a

prime number appearing, then A's

complement is the event of a non-prime number appearing.

If A is an

odd number, then A's complement is an

even number. Obviously, the complement is the

opposite event. So,

mathematically, the formula for the probability of the complement

ee of an event A PA^ C is equal

to 1 - the probability of A. For more

details, pay attention to the following example.

The chance of someone being accepted into their dream state university

is 0.54.

What is the chance that he will not be accepted into

that PTN? This is an opportunity to be accepted,

right? This opportunity is accepted. This is the PA

who asked about the opportunity not

to be accepted and the opposite opportunity. This means that what is

being asked here is the probability of

the complement. So if the PA is

0.54,

then what is the probability of not being accepted or the probability of

A being the complement of 1 - PA 1 - 0.54

? 0.46. This is a chance he won't be

accepted. Okay, another example. Okay,

now let's discuss the second example. Seven

people sat around a round table.

What is the probability that three particular people

will not be seated next to each other? Okay, let's

answer. For example, event A is when

three people are always side by side.

While here the questioned

will not sit side by side. That means the

opposite happened, right? So, for

example, if event A is three people who are

always side by side, then when they are not

side by side, that is the

complement, right? Well, now I will

calculate event A first. When three

people are always side by side, we use

ee cyclical permutation, yes. There are

seven people here. So, for example, if this is a

round table, then these seven people

, three people are always

side by side.

For example, these three people,

they are always side by side. We consider this

as one element, yes. So

how many are there here? 1 2 3 4 5. So here we

consider these as five elements. Now

we use cyclic permutations. Remember the

cyclic permutation formula is the same as n

- 1 factorial, right? We have

learned this in the rules of enumeration. So

if there are 5 elements then NA = 5 - 1

factorial. But these three people who are

always side by side can move,

can move from each other, but remain

side by side. In how many ways? With

3 factorial ways. So we multiply this

by 3 factorial. 5 - 1 is 4. 4

factorial * 3 factorial. 4 factorial is

24. 3 factorial is 6. 24 * 6 = 144.

Now, for those of you who are still confused

about this material, you should

first study the cyclic permutation material.

I included the link in the description of this video.

Well, now we will calculate NS. Since

there are many possibilities for

calculating NS, these are not always

contiguous. So,

how many people do we count? There are 7uh people. So,

n is 7. So, ns is equal to 7

- 1 factorial. This is the cyclic permutation formula

. 7 - 1 is 6. 6 factorial is

720.

So the probability of 3 people always being side by side

is 144

/ 720

what? That's the same as 1/5, right?

Well, this is just the chance for 3 people to always be

side by side. Meanwhile, what is being asked

is the probability of three people not sitting

next to each other. Now we look for

the complement. The probability of 3 people not being

side by side is PA complement = 1 -

1/5 what is? 4/5. This is the opportunity.

Okay, now we move on to the next sub-topic,

namely about

expected frequency. The formula for expected frequency is

simple. The expected frequency of an

event A is the probability of A times the number of

trials or the number of events.

For example, if two dice are thrown

simultaneously 72 times, what is the

expected frequency of the sum of the two dice being

more than or equal to 10? More than or

equal to 10. Okay, now let's

solve it. Ee, the possibility of the total of the

dice being more than or equal to 10. This means the

total could be 10, 11, or

12. So if we

write the possibility when the total is 10, the

first die could be 4, the second die could be

6. This total is 10. Or the

first die could be 5, the second die could be 5 too. Or the

first die is 6, the second die is 4. This

total is 10. Or maybe the

total is 11, namely the first die is 5, the

second die is 6. Or vice versa, the first die is 6, the second die is 5.

This total is 11. or the

last possibility is the total is 12. The

first die and the second die are both

6. Okay, so what is the NA here?

1 2 3 4 5 6. There are 6 possibilities for NA.

For NS, one die has 6 sides,

right? If two dice means 6², there are

36. This is the NS. So the probability is

6/36 or we simplify it to 1/6.

Well, now we calculate the

expected frequency. The expected frequency is the

probability of event A times the number of

trials. Here the number of trials is

72 times. So the expected frequency

is 1/6 * 72,

right? 72 / 6 is 12. 1 * 12.

Well, this is the expected frequency. Well, this

means that out of 72 throws, there are

12 throws where the

total of the two dice will be greater than or equal

to 10. The second example, based on the

weather forecast, the probability of it not

raining in Tasikmaya City during

November 2020 is 7/15.

How many days is the expected rain in

Tasikmaya City during November

2020? ee 7/15 there's a chance it won't

rain. There's no chance of

rain here. So, for example, event A is an

event where it doesn't rain. So

7/15 is the PA. The chance of no

rain is 7/15.

So the chance of rain is the

complement of the opposite event, right?

So the complement PA or probability of

rain is 1 - 7/15.

1. Friends, just change it to 15/15

then subtract 7/15 to get

8/15. Well, this is the chance of rain or

complementary PA. Well, now we calculate the

expected frequency ee. What is the

expected frequency of rain in

Tasikmaya City during November? The

number of days in November is

30 days. So the expected frequency

is 8/15 * 30. 30 / 15 is 2. 2 * 8 =

16. So the expected frequency is 16

days.

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Okay, that's it for this video.

See you in the next video.

Asalamualaikum warahmatullahi

wabarakatuh.