Introduction to Probability

In today's chapter we will see what probability is.

We have been learning probability since our primary class.

Probability means what is the possibility of an occurrence of any event, which means what is the likelihood of an event to happen.

We calculate this in between zero and one and we use one formula to calculate it which is as follows, probability formula of any “a” event is number of favourable outcomes upon total number of possible outcomes, which means the total outcomes and that I have with me, if I divide that with the cases possible in that particular event then I will get the probability of “a” event.

So, suppose, I will take one example, I have one coin, I will toss that coin, I have two possible outcomes, either on tossing I will get heads or I will get tails on tossing.

Now, if I get heads then what will be my total possible outcomes, heads or tails, means my total possible outcomes are two, the possibility of getting heads is 1/2 because number of chances 1 can come, it can come once upon 2.

So, the probability that heads will come after tossing one coin that becomes 1/2.

I will take one more example.

If I roll one dice, what will happen after rolling the dice? In there the total number of outcomes that I have are six.

I can get one after rolling it, I can get 2, I can get 3, 4, 5 or 6.

If any value comes from these 6 from that my total outcome becomes 6.

But if someone tells me that I want to know the probability that 6 should come, what the probability of getting 6 alone is, it can come only once.

So, my probability that I will get six on my dice, it's probability becomes 1/6.

Before seeing the probability ahead, let's understand what are the different types of it.

Like I have two types of probabilities.

One is mutually exclusive and the other is non-mutually exclusive.

Now, what are these? Let's see it once.

Mutually exclusive means suppose I have one event, there is one possibility that I have tossed one coin, after tossing that coin, either head will come or tails but both those events cannot come at one time.

This means that such an event where there is the possibility of getting only one outcome, we call it a mutually exclusive event.

Suppose, after rolling one dice, even after rolling the dice once, I can get 1 number or I can get 6 on the dice or 5 can come on that dice.

But, together 1 and 6 cannot come on that dice, that's why we call it a mutually exclusive event.

What is the meaning of non-mutually exclusive? Suppose I have a stack of cards.

From those playing cards I will pick out one heart and one king, which means I have to pick one K and one heart card.

It can so happen that the card that I've picked is of King, it is possible that it can be a heart of king card or it is possible that it can be any other card of hearts.

So, what happens in these three possibilities? We have picked one card but the two outcomes that have come are possible together in this scenario.

So, we will call this a non-mutually exclusive event.

We will see it once.

Since what happens in mutually exclusive events.

Both my events are independent from each other, they are not crossing each other.

So, my Venn Diagram of these both would be like an independent event where A is a different value and B is a different value and they are not colliding with each other.

But what is the meaning of non-mutually exclusive? It means that I have one event of getting a king and one event of getting a heart and there is also one more possibility that I will get king of hearts, that we have marked with this purple colour where the values are coinciding.

Now we have understood what is mutually exclusive and what is non-mutually exclusive.

But further how do we have to calculate the probability? We have few rules for it.

Like additive rule of probability or multiplicative rule of probability.

We will understand it through an example.

Suppose, I have conducted an experiment where I have rolled one six-sided dice.

Now someone asked me the probability of getting two or five on my dice, how much is the probability of getting them.

Now you pay attention here.

This will be one mutually exclusive event.

Why? Because either two can come on my dice or five can come on the dice.

 Two and five cannot come together.

So, we will call it a mutually exclusive event and we will calculate it for the additive rule of probability.

So, to calculate it, I have got two events, probability of A or B.

We will calculate it through sum, which is probability of A plus probability of B.

Which means if I have to calculate the probability of 2 and 5.

I will have to first know the probability of 2 and I have to know the probability of 5.

2’s probability would be 1/6.

Probability of getting 5 would be 1/6.

We added them both.

It is 2/6.

So, my total probability is 1/3.

Next, we will take one more example to understand the additive rule of probability.

The previous example that we had seen was for mutually exclusive.

Suppose we have removed one single card randomly from the stack of cards, in that I have to find the probability whether it is a king or is it a club? Okay.

Now in this particular scenario like we had discussed, this is a non-mutually exclusive event.

Why? Because the king of clubs is repeated twice, one is if I take out the king on its own, then it will be calculated and even when any card of clubs comes.

It is a possibility that it is the king of clubs.

 So, in this way that particular probability which we are calculating extra, we can get that subtracted.

So, in this case, my additive rule of probability gets a little modified, it changes a little.

Now, we will see the probability of A or B.

In the previous case we were seeing probability of A plus probability of B but now since we have got one additional value so we will subtract that value.

Which means now my value will be probability of A plus probability of B minus probability of A and B.

If I put these values in this example.

Then my probability of king or club will convert into probability of king or probability of clubs, form that I have subtracted king of clubs.

The probability of picking king from my stack of cards would be 4/52.

Why? Because the total number of cards that I have are 52 and I have four kings.

So, my probability of King would be 4/52.

The probability of getting a club would be 13/52.

Which means from 13, I can get any card of clubs and 52 are my total number of cards.

But King of Clubs would be only one card with me from the total 52 cards, which I have subtracted.

After doing this calculation, the probability that I get a king or club is 4/13.

So, in this same way, we see two types of additive rule of probability.

One is for mutually exclusive events and the other is for non-mutually exclusive events.

What we saw was mutually exclusive? If I want to take out the probability of A and B.

I will calculate both probability of A and probability of B and I'll add them up but what will happen in non-mutually case? I will have to take out for A as well as for B and I will subtract the common event from it.

Now we will see the multiplicative rule of probability.

This is a very important rule.

Why? Because many times it happens.

We don't pick out the card from the deck only once, many times it so happens that I have not rolled the dice only once.

It can happen that I have tossed my coin thrice.

So, what happens in that particular scenario? I have to use the multiplicative rule of probability, before understanding it let's understand what can be two particular events, like the first event is an independent event.

Independent, like its name suggests, suppose, let’s consider that I have tossed one coin thrice, after tossing it thrice I am getting heads the first time, second time I get tails and third times heads.

But if you see all three, my probability of getting heads is also ½, my probability of getting tails is also ½.

And the probability of getting tails the third time is also ½.

So, this means that my event in which the number of outcomes are not reducing.

 Which means 1/2 is my probability in every event, in every outcome.

So, we call that particular event an independent event.

What happens in this case? We calculate every event’s probability separately, we calculate each one's probability separately and we multiply them.

The final value that I get, comes out to be the value of an independent event.

Now, there is one other event dependent event.

This means that if my second event outcome is getting affected by my first event outcome, because of which my second event’s probability changes, means my number of outcomes starts reducing in my other events then we call it a dependent event.

Why? Because one value depends on the other value.

Let's understand this with an example.

Suppose, from the deck of 52 cards I have taken out three cards.

Okay.

And I have taken those three cards one by one but not placed them back in the deck.

This means that I have kept them without replacing, this is the probability example that we are seeing.

Which means in this particular scenario, I have to remove the probability, what is the probability of three A's if I have to remove three A's one by one in three cases.

Now, the first scenario will be that I can have 4 A’s from the 52 cards.

So, in my first pick, the possibility of picking is 4/52.

Now, since I've already taken out one card, I have not replaced it.

So, the total outcomes that are left with me or total cards left with me are 51 So, now if I am picking out another card, it's probability with me is 3/51.

Why? Because one A has already been taken out.

So now I've got 51 cards.

And since already one A has gone, so now there are three A's left.

In the same way if I'm taking out this A in the third round, now I have got only 50 cards left.

Why? Because I've already taken out two A's.

So, in this way the 3rd case’s probability or the probability of taking out the third A is 2/50.

In this scenario when we are seeing that one by one my second event’s outcome is dependent upon the outcome of the first event.

If I would have kept that card back then all the time, I would have 52 cards in my deck of cards.

My outcome would not have changed.

This probability or this event is called a dependent event.

We can see through two experiments how multiplicative rule works on dependent and independent.

Suppose I have done the first experiment where I have tossed one coin and then in another event, I rolled one dice.

So, I have to tell them what is the probability of getting heads on my coin and along with that the probability of getting 3 on my dice.

Now we will see here that both my events are independent.

Why?  Because whichever value comes on the coin, it will not affect my dice.

Whatever value comes on dice, it will not affect my coin.

Since they are independent events, we will calculate their respective probabilities and multiply them.

Which means probability of A and B will be probability of A multiplied by probability of B.

Which means that the probability of head and three together.

What will be the probability of head? ½.

And the probability of getting three would be 1/6.

So, my total probability would be one by 1/12.

Now we will take another example, in that example, suppose from a deck of 52 cards, I've taken out one card and after removing that card, I didn't put it back but I drew another card from that deck.

Basically, from one deck of cards, I have taken out two cards.

So, I want to know that my first card that I have taken out should be queen and the second card that I have taken out should be Jack.

I want to see the probability of both of them coming together.

It is the same scenario which we just saw.

Since they are dependent events, in this particular case, what is it? My multiplicative rule’s formula is a little different.

Now my P of A and B will become the probability of A multiplied by probability of B upon A.

Here B upon A is called the condition probability.

This means that probability of event B when A has already happened, which means that my probability of taking out Jack when I've already taken out the Queen's card from it.

This is our one important term, conditional probability.

We will study it further in data science, in the Naïve Bayes theorem and they are used in several places.

So, pay attention to this concept of conditional probability.

Now let's come back to this example.

I have to take out the probability in the first pick of the queen and in the second pick of the jack, even though my first card was Q.

So, in the first pick what will be my probability, that will be 4/52.

Because I have 4 queens from the total cards.

So, 4/52 would be my queen’s probability.

Already one card has been removed from it.

What will be my J’s probability? 4/51.

Why? Because I have 4 J’s with me and the total outcomes left with me are 51.

Queen and Jack total probability will be 4/52 X 4/51.

So, in this way we use the rule of multiplicative probability and we calculate what would be a different outcome’s probability.

Now in this chapter we saw what is the probability, what are the additive rules, how can we use multiplicative rules.

How we calculate probabilities for mutually and non-mutually events.

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