As quantum mechanics is probabilistic in nature.
We should have a basic knowhow of probability and how mathematically we calculate probability.
So, for example, I'm tossing a dice, which is a six phased dice, and if I want to find the probability
of getting three.
The answer is one by six.
Six is the total number of outcomes, and we want to predict three.
So the answer is one by six.
In our day to day life.
We perform many activities which have a fixed result no matter any number of games, they are repeated,
for example, given any triangle without knowing the three angles, we can definitely see that the sum
of measure of angles is 180 degree.
But in cases like tossing a coin or dice, we cannot tell a result in advance.
These are called random experiments, so an experiment is called a random experiment.
If.
It has more than one outcome.
And.
If it is not possible to predict the outcome in advance.
In this video of probability, we shall refer the random experiment as an experiment.
So we should note two things in probability.
One is a sample space.
And another is an event event is something that we want to predict, whose probability we want to predict,
that is the outcome or the outcomes chance we want to predict.
And sample space is nothing but the total number of outcomes possible.
So if we toss two coins, the set of possible outcomes can be.
Heads, heads.
Heads still.
Bill heads.
Or Ortel Bill.
So these are the list of possible outcomes, hence it's called sample space.
Now let's see if we want to find the outcome, having exactly one head from the list of sample spaces.
Yeah, so that we want to predict.
So the event is HP and be it.
Here are some other algebraic operations that we should know on asset.
So it was ample space of a ruling.
This is one, two, three, four, five, six.
Let's say the dice is six viscid right to the total number of possible outcomes could be from one to
six.
Leota is an even.
Yeah, having outcome two, three and five, B is equal to one three five.
So over here, A Union B is equal to one, two, three, five.
A intersection B is three commodify.
A minus B is equal to two.
And a complement is equal to one four six.
Now, what are exclusive events, let's say the sample spaces one to six, it is a set of odd numbers
in their dyce.
B is the set of even numbers in today's.
If A Intersection B.
Is equal to null said that there is no intersection at all.
Then both are disjoint sets.
So both A and B over here.
Are exclusive events because they don't have any intersection with each other.
So.
Let's take a very, very simple example of a six facing days, what is the probability, your chance
of getting one?
The answer is one by six because the sample space is one to six.
And even the outcome to predict is one, which is a single event.
So the probability is one by six.
Next question, what is the probability that the outcome would be in order no.
Yeah, so if it's all from one to six, which are the odd numbers, one, three and five.
Samples based on total possible outcomes are one to six, so the answer is possible outcome divided
by the total possible outcome that is three by six is equal to one by two.
Yeah, so the answer for the probability of having an odd number in a six facer dies is one by two.
Now there is something called as conditional probability, what exactly do you mean by conditional probability?
It means that a particular event is given to you, a particular condition is given to you.
Based on that event, condition constrained.
Find the probability of another event, so.
I have given you a constraint or a given condition from that, find out.
The probability of an event, so, you know, it becomes a subset of a sample space.
Based on the condition.
So let's say we have two events, event and even B and the outcome of event, it can affect the probability
outcome of B.
So this is referred to as conditional probability, here is a very simple example of conditional probability.
Let's say if I want to buy something which could create value.
With which I would be happy, but the given condition is there is a budget assigned to me.
So within that budget, I got to buy something.
Yeah, so that's called condition, yeah.
Let's check out an example of conditional probability, let's say we are about to toss three coins so
the sample space would be.
Heads, heads, heads.
Heads, Bill, heads, heads, heads.
Bill.
Bail, bail, his bail, bail, bail.
Bill heads Bill.
Bill heads, heads or heads bail bill.
Lizzi, based on the Abbo sample space, that is the total number of outcomes possible event is given,
what is even the event is having exactly two heads.
Yep, so that's a constraint which is given and we need to predict the probability of event B having
the first coin as Hartill.
What should be the answer in this case, there is a very fascinating theorem called base theorem based
theorem comes pretty handy in solving conditional probability.
So here we can solve using this equation.
This equation can be pronounced as probability of be given a.
Is the constraint is equal to probability of A intersection, B, divided by probability of A.
So how conditional probability is calculated?
Here is a very simple picture diagram which will tell us how a conditional probability is calculated.
So let's say you have probability in one direction and probability a compliment in another direction.
Right.
And if I want to find out probability of being given a.
It can be given us probability of a intersection, B divide by probability of it, if I want to find
out probability of B compliment given here, the answer is probability of A intersection, B, complement,
divide by probability of it.
If I want to find out probability of being given a compliment, the answer is probability of a compliment.
Intersection B divide by probability of a compliment.
Similarly, probability of big compliment given a compliment.
The answer is probability of a compliment, intersection B compliment, divide by probability of a compliment.
So the answer.
For the above problem statement is.
We know even here exactly two heads, so the probability of it is equal to three by eight, eight is
the total number of outcomes our sample space be.
Is what?
Forestdale, the first coin should be Ortel, so probability of BES four by eight, which is equal to
one way to calculate the answer based on based theorem, we need to have a probability of a intersection
B, which is one by eight, and the answer is probability of A intersection B divided by probability
of error is equal to one by three.
I hope now the base theorem and the conditional probability is clear.
Cheers.