Hello everyone, welcome back to a new video.
The topic we are going to cover in today's video is a
probability topic.
All of you must have at least studied the probability topic
till 10th standard.
Be it science students or commerce or arts background
students, whoever is from the background or from the
stream, they must have at least studied the probability
topic till 10th standard.
And the probability scoring used to be till 10th standard.
So, even now our probability is not that hard.
We will see the basic formulas, we will see the terms used
in it, we will see the cases in probability, and then we
will do their questions on their basis.
So, let's see.
First of all, we will understand what is probability.
So, the simple meaning of probability is the probability of
anything happening or not happening.
It means the certainty of occurrence of any event.
The certainty or surety of any event happening or not
happening.
That is what we call as probability in simple definition.
The most basic example in probability is
Toss of a coin.
We flip a coin.
As soon as we toss a coin, we have only two options.
Either you will get a head or you will get a tail.
So, what we have to find out is what is the probability of
getting a head?
What is the probability that if we toss a coin, we will get
a head?
Or what is the probability that we will get a tail?
So, how do we find out this?
We will see that.
First of all, you have understood the definition of
probability.
Then, you should know the next case.
You should know the next term.
That is sample space.
You might not know this.
As such, you might know.
For example, I gave you an example.
You toss a coin.
So, all the possible outcomes that you have, they come in
your sample space.
All the possible outcomes that you have, like if we are
tossing a coin, what are the possible outcomes that we have?
Either we will get a head.
One is a head.
And what is the other one?
You will get a tail.
There are two possible outcomes.
So, how many elements are there in your sample space?
You have two elements.
What is sample space?
It is a set of things.
It is a collection of things.
All the possible outcomes that we have can be of any random
experiment.
Whatever we are doing, like tossing a coin, or looking at
the case of cards, or looking at the case of dice, all
these are called random experiments.
Then, the cases that are given to us, like we are given
that we have to find the probability that we will get a
head.
What are you doing?
You are getting a head.
What are you getting?
You are getting a head.
So, in this sample space, you are given a case.
So, what is this case called?
It is called an event.
This case is called an event.
And on whose basis is this?
Whose sub-case is this?
It is a case of tossing a coin.
So, tossing of a coin, what is this? This is known as a
random experiment.
So, if we perform a random experiment, and if we are asked
the probability of any event, then we will find out the
probability.
Now, these were the main terms that you should know so that
you get clarity in reading the question.
The next main formula is the main formula of probability,
which we have been studying since childhood.
See, what is the probability of?
We will find out the probability of getting a head.
And what is this?
It is an event.
So, what is the probability of?
We will find out the probability of any event given to us.
So, the probability of any event, any event, so, to find
out the probability, we will write P.
Here, we will write the event, whatever event is given to
us.
For example, what was the event?
Head.
In brackets, we will write head.
Or simply,
P.
In brackets, we will write E.
And you will have to mention E above.
Which event do we have?
Simply.
So, what is the probability of any event?
First of all, the number of favourable favourable
favourable favourable outcomes.
Number of favourable outcomes.
Now, what are favourable outcomes?
For example, you asked about head.
So, which option is there in favour of head?
There is only one option, head.
So, those are your favourable outcomes.
Now, how will we divide it?
We will divide it by total total number of total number of
total number of total number of possible outcomes.
What are the possible possible outcomes in this experiment?
We will divide it by simple example.
For example,
You have performed an experiment.
You are tossing a coin.
Now, this is a fair coin.
This is what we see in our life.
If we toss a coin, it can be any coin.
It is not unfair.
Nobody has any advantage.
If we are tossing a coin, you are asked to find the
probability of getting a head.
to find the probability of getting a head.
Find the probability P
I will write head in the bracket so that we can find the
probability of getting a head.
The probability of getting a head.
First of all, we will find out its sample space.
In the sample space, the total number of possible outcomes
is denoted by
S.
is denoted by S.
What are the elements in S?
First of all, either you can get a head or you can get a
tail.
These are the two possible outcomes in tossing a coin.
The event given to you is getting a head.
To find out the probability of getting a head, we have to
find out the probability of getting a head.
getting a head.
What are the favourable outcomes of tossing a coin?
If you have only one element, then you will get a set of
all the possible outcomes.
We always keep the set in curly brackets.
Getting a head is the only possible outcome of tossing a
coin.
Getting a head is the only possible outcome of tossing a
coin.
How many members are there in E? There is only one member
in our event.
That is head only.
How many elements are there in the sample space?
There are two elements in the sample space.
Two elements.
All the elements in the sample space are denoted by S.
N denotes cardinality or the number of elements in the
sample space.
To denote this, how many elements are there in the sample
space?
We will denote it by
NS. How many elements are there in the sample space?
Two.
NS is equal to 2.
Similarly, how many elements are there in E? We will denote
it by
NE.
How many elements are there in E? There is only one element
in E.
So, the value of NE is 1.
Now put these values in the formula.
What will be the number of favorable outcomes?
NE is the number of favorable outcomes of the event divided
by the total number of outcomes.
That is, how many elements are there in the sample space?
NE is equal to 1.
The total number of favorable outcomes is 2.
half that is 50 percent if we take out half in percentage
then what is the probability of you getting a head?
Probability is half and if you want to say it in percentage
then there is a 50 percent chance that you will get a head
in TOSS.
Similarly, what is the probability of getting a tail? You
will have a 50 percent chance of getting a tail.
This is one example.
I will give you the second example.
In today's time, we will address the main important cases.
Then in the next video, we will do question on the
probability.
Okay? So, first of all, you came across tossing of a coin.
This is your first experiment.
So, for tossing of a coin, we just saw that we have two
elements in the sample space that is head and tail.
And even if you are asked, you can find out how many
elements are there in the sample space.
So, this is our first random experiment that is tossing of
a coin.
Our second random experiment will be that we are not
tossing a coin once, but we are tossing a coin
simultaneously twice.
How? Either you say that you are tossing a coin twice.
So, you will get the outcome of the first one.
Then you will get the outcome of the second toss.
So, your outcome will be in pairs.
Either you can say it like this or you can say that you are
tossing two coins.
It will be the same thing that you should get two outcomes
in pairs.
So, if we are tossing two coins, then what elements will
you get? Either you will get head in the first one and head
in the second one.
Or you will get head in the first coin and tail in the
second one.
So, the second outcome will be that you will get head in
the first one and tail in the second one.
Then what will you get? You will get tail in the first one
and tail in the second one.
Then next you will get tail in the first one and head in
the second one.
Apart from this, you are not getting any other possibility.
So, when you were tossing one coin, how many elements were
there
in your sample space? Only two.
When you were tossing two coins, how many elements were
there in your sample space? Four.
How can you write four? Two power two.
Similarly, if you are tossing three coins, what will happen
in that? Two power three will be there in the sample space.
Similarly, whenever you are tossing one coin, how many
elements will be there in its sample space? Two power n
elements will be there in its sample space.
Two power n elements.
There are so many possible outcomes to toss a coin.
Here, what is n?
Number of times coin is being tossed.
coin toss or number of coins we can call it indirectly so
it will give you number of elements in sample space in case
of coin it will give you formula that is 2 to the power n
so this is coin case next case which is foremost important
that is dice rolling
of a dice everyone must have played Ludo in childhood must
have played business in all you roll a dice and how many
outcomes can come out of it how many are there 1 to 6 1, 2,
3, 4, 5 and 6 so if you roll a dice if you roll a dice so
if we see case of
dice so you have first experiment of tossing a coin second
experiment that is rolling of a die rolling of a die
rolling of a dice so how many elements will come in sample
space if we take out S so either 1, 2, 3 4, 5 or 6 these
elements can come these
numbers will show if we roll a die so in this sample space
how many elements are there in total in this sample space
1, 2, 3, 4, 5, 6 total 6 number of elements now on this
basis it can ask you any question let's suppose it is
asking you this is your
experiment you are rolling a die if you are given an event
that probability of getting an even number that you get an
even number your answer will come that number will show
that number is an even number we have to know its
probability so first of all
how many favourable outcomes can we get so what are the
favourable outcomes in this set even number 2, 4, 6 so we
will mention all of them 2, 4, 6 is there any other option
no other option no favourable outcome so our n e value will
be how many elements
are there in e 3 elements so probability of e getting an
even number will be n e by n s means favourable outcome
upon total number of possible outcomes n e is 3 and divided
by n s is 6 so you have 1 by 2 so this is also a
possibility similarly, there
can be other cases getting an odd number or getting a prime
number or getting a square which is a perfect square so
according to that e will be same in all the questions if
you have a dice case 6 elements will be there if we are
talking about fair die
and e will be the event will change as per the demand of
the question we will see the event and then we will put the
formula and get the value so this was your second important
that is dice case now as I said in coin if you pick up the
coin once or you
pick up the coin twice or three times or three coins
similarly, what will happen in that case next case suppose
you are rolling two dice what? rolling of two dice if you
are rolling two dice so what will happen in the first case
6 elements will be there
in the first die how many possible outcomes will be there
in the first die 6 or 1, 2, 3, 4, 5, 6 or 1, 2, 3, 4, 5, 6
similarly in the second die also 6 cases will be there 1,
2, 3, 4, 5, 6 1, 2, 3, 4, 5, 6 now if 1 is there so
corresponding in the second
die 1, 2, 3, 4, 5, 6 1 can be paired up with 6 similarly 2
if 2 is there in the first die so in the second die also 1
to 6 can be paired up 2 will also be paired up with 6 3
will also be paired up with 6 6 x 6?
What will we do with 6 x 6?
We will pair up.
So in this your sample space becomes 6 x 6 that is 36
elements.
Sorry this is NS.
If you are getting how many elements in sample space, how
many possible outcomes can be there for this experiment?
How many will be there? 6 x 6 that is 36 cases.
And how will they be?
See if I make a sample space, then you will have that.
First I will write the pair of the first die and second I
will write the element of the second die.
So the first pair will be 1.
If 1 is coming, then anything can come from the second die.
1 can come.
Then 4.
4 can come.
Then 5.
5 can come.
Then 6.
6 can come.
Then 7.
7 can come.
Then 8.
8 can come.
Then 9.
9 can come.
Then 10.
10 can come.
Then 11.
11 can come.
Then 12.
12 can come.
Then 13.
13 can come.
Then 14.
14 can come.
Then 15.
15 can come.
Then 16.
16 can come.
Then 17.
17 can come.
Then 18.
18 can come.
Then 19.
19 can come.
Then 20.
20 can come.
Then 21.
21 can come.
Then 22.
22 can come.
Then 23.
23 can come.
Then 24.
24 can come.
Then 25.
25 can come.
Then 26.
26 can come.
Then 27.
27 can come.
Then 28.
28 can come.
Then 30.
30 can come.
Then 31.
31 can come.
Then 32.
32 can come.
Then 33.
33 can come.
Then 34.
34 can come.
Then 35.
35 can come.
Then 36.
36 can come.
Then 37.
37 can come.
Then 38.
38 can come.
Then 39.
39 can come.
Then 40.
40 can come.
Then 41.
41 can come.
Then 42.
42 can come.
Then 43.
43 can come.
Then 44.
44 can come.
Then 55.
55 can come.
Then 62.
62 can come.
Then 63.
63 can come.
Then 64.
64 can come.
Then 67.
67 can come.
Then 68.
68 can come.
Then 70.
70 can come.
Then 71.
71 can come.
Then 72.
72 can come.
Then 73.
73 can come.
Then 74.
700 can come.
7 Forgive me.
I can't think of 19.
2,000.
No problem.
No problem at all.
7 times as much as you want to.
3rd important example is of cards.
But before discussing about cards, let us see the
properties of cards.
So, what are the probabilities?
In layman language, we generally talk in percentage.
We understand better to compare in percentage.
For example, if someone says that you will meet me
tomorrow, what is the percentage that you think you can
come or not?
Will you come or not?
If a person says that he will not come, he completely
rejects.
He directly says that he will not come.
So, if he does not come, then it is a sure event that he is
not coming.
So, there is zero probability.
What is the probability of him coming? Zero.
If he is not coming, then what is the probability of him
coming? Zero.
So, the value of probability starts from zero.
Now, there will be zero probability.
For example, if a person is not coming, then what is the
probability of him coming? Zero.
Similarly, if a person says that he has a 50% chance of
coming, then what will happen?
If you change 50% into a fraction, then it will become 1 by
2.
Similarly, if a person says that he is 80% sure that he
will come, but if 20% of his family members come, then he
may not be able to come.
So, if you change 80% into a fraction, then you will get 4
by 5.
Now, the 4 by 5 value is greater than 1 or less than 1?
Obviously, the number above is less than 1 and the
denominator below is greater than 5.
If you divide, then the value you will get is less than 1.
So, if you go from 0% to 99%, then your probability will
always be less than 1 and greater than 0 or equal to 0.
So, the probability can be greater than 0 or equal to 0.
So, greater than or equal to 0.
And your probability will always be less than 1.
Now, if your friend says that he is 100% sure that he will
come tomorrow, then you will divide 100% into a fraction.
100 divided by 100 is 1 by 100.
So, 100 divided by 100 is 1.
So, when you have a 100% sure event, then your probability
of that event is equal to 1.
So, this is the case of equal to.
So, this is the first property of probability.
The probability of any event, whether it is any event in
the world, will always be equal to 0 or greater than 0 or
less than 1 or equal to 1.
So, what is the range of our probability?
It is the range of probability between 0 and 1.
So, the first case is the range of probability of an event.
Now, if your probability is equal to 1, then you will call
it a sure event.
Now, the second case that I am going to tell you is,
suppose you have only two choices in any event.
There can be two or more choices.
But, if you divide that case into two, like you tossed a
coin, then you had two choices.
One was either head or tail.
So, if I call one event as getting a head, and if I ask you
the opposite of that event, then you will read this in sets.
If I ask you the opposite of getting a head, then you will
call it a compliment.
In the language of Maths, it is called a compliment.
So, if you get everything except the head, then you will
call it the opposite of that event.
So, if I am denoting E for event, then I will denote E for
compliment.
C for compliment is a small c.
E for compliment is a small c.
Or, you can put a dash on E.
These two represent compliments of any event.
Now, if we have E for getting a head, then what will be the
opposite of that?
We should not get a head.
Right? Not getting a head.
So, if you have E for getting a head, then E for compliment
is not getting a head.
Not getting a head means there is only one option left.
So, if you have E for any event, then what will be the
probability of it? I will denote it as P.E.
So, what will be the probability of its compliment?
It will be 1 minus P.E.
What will be the probability of any event's compliment?
Minus P.E. from total.
What is total?
Probability of sample space, which is 1.
Total will be 1.
So, 1 minus P.E. is equal to P.E. compliment.
Probability of any compliment is 1 minus P.E.
This is the second property.
Now, third.
This is not the third probability.
This is a formula.
This was a small example.
Probability of getting a 1 in a dice
Probability of not getting a 1
Probability of getting a 1 in a dice
Probability of not getting a 1 in a dice
Probability of getting a 1 in a dice
Common element in both the dice is 2.
Common element in both the dice is 3.
Common element in both the dice is 2.
1 so 1-1 by 6 which is 5 by 6 and if you look at the
inter-union one, what is the probability of this?
This will also be 5 divided by sample space that is 5 by 6
so you can also get it from the formula or if you directly
know the values of all, then you can also get it from it.
So I hope you will be able to understand the formula which
question we are
using.
Next, the cases that will come in this word, one of them is
independent event.
Independent event, mutually exclusive event, exhaustive
event, we will do that on the basis of questions.
But let me tell you what is independent event?
When both the events are not related to each other, like in
this case, there was a relation between these two that 2
and 2 was common.
Independent event, both will run separately in their own
way, they will not have any role with each other as such.
So, we will see what we use for these words and we will
also see what we use for these words.
So, we will see this on the basis of questions and I will
explain to you as and when the questions come.
Now, we have done the properties, we have done the main
cases of two examples, that is of coin and dice.
The third important example is of cards, playing cards, you
must have seen that how many cards do you have in total,
there are 52 cards, in 52 you have two categories, that is
why you will have two categories, one is of red color and
the other one is
of black color.
So, one is of black color, that is of this shape and the
other one is of this shape, okay, I will write the names of
these.
And the second one is of red color category, that is of
this shape and the other one is of this shape.
so total you have 4 categories but if you see color wise
there are 2 categories one is black and other is red so in
these 4 categories you have individually 13 members if I
write their names so this is your diamond card this is your
heart card this is
your club card
and this is your spade card which is hukum ka patta spade
card so these are the four names of these different
categories now how many elements are there in total? 13
there are 13 cards in this also there are 13 cards in this
also there are 13 cards in
this also then if you add up all these, then you will get
total 52 cards now what are those cards? there are only one
type of cards in sum that is, one you will have that is ace
card one you will have that is queen one you will have that
is king and one
you will have which is a joker not joker, jack you can see,
it is not joker it is jack
so these are the face cards because they have faces next
you have four main cards then you have cards from 1 to 10
so 10 cards are zero one is called ace so you will remember
it in ace category so from 2 to 10, you will have this
number one is called
ace so you will have 8 and 4 total 8 plus sorry, this is 9
and 9 plus 4 you will have total 13 cards in one category
so all the categories will be like this in all the four
categories so you will have 52 categories so this was the
overview of cards if
someone has never played then this is for them now we will
do the questions in next class on their basis first we will
do easy questions and then we will do tough questions in
which you will find the use we will see those questions
that we can do same
type of questions with permutation and combination of
probability
I hope you have read permutation and combination in this
video and if you have not read then please watch the video
on permutation and combination before watching the question
video so see you in next video in question video thank you
for watching the
video
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