Sometimes things get even worse.

You don't have data, you have no clue

and at that point we go to the subjective

probability definition.

Very important, for example, if you have a new product development

you bring out a brand new product, really disruptive innovation

you have no idea whether your customers like it or not.

Maybe they have some, what managers like to call it, gut feeling

or experience.

And based on that experience, you say: "Oh, I think there's 75%

chance that will be a successful product".

And, so in that case we talk about subjective probabilities.

Very important in managerial decision making

and everyday decision making.

Now, basic probabilities have to satisfy some goods.

Now, I have to give you 3 rules, looks rather mathematical

there's nothing to be understood here.

These are also called axioms.

Or after Russian mathematician who was the first to write these down

in 1933, also called Kolmogorov's axioms.

These are just assumptions that we put in place

for probability theory.

Everything else can be derived from this.

Rule #1.

The probability of any outcome in sample space P of S must be 1.

Something must happen when we do our random experiment.

P of S is 1.

Second rule, very intuitive, any probability is a number

between 0 and 1.

Can't be larger than 1, cannot be negative.

Please keep that in mind.

And finally, the first little more complicated rule

if you have 2 events that have no elements in common

also called disjoint by mathematicians

the end of probability that A or B happens

or the probability that A union B happens

equals the sum of the individual probabilities P of A + P of B.

If that looks a little tricky already to you,

let's look at an example and let's go back to Fair Die.

What's random experiment?

I'm rolling a fair die.

I don't know what's going to happen.

That's now my experiment.

What are the possible outcomes?

1, 2, 3, 4, 5, 6.

And they together build the sample space.

Now, what I told you about events?

Events are subsets of S, and here are defined

I pick an event A, the even numbers

2, 4, 6 and the event B 1 and 5.

And now let's look at A and B and their probabilities

and see how this works.

What's the appropriate definition here?

I don't have to make up probabilities.

There's no subjective probability here.

I don't have to roll the die 1000 times to determine

empirical probabilities.

Since we believe it's a fair die, we can use a classical definition

All 6 numbers are equally likely.

And so, I'm allowed to just divide.

So, clearly P of any number 1 through 6

is equal to 1.

Seven cannot happen, zero cannot happen, pi cannot happen.

Now, let's look at the probabilities of the 2 events.

A has three elements - 2, 4, 6

3 out of 6 is a 1/2 = 0.5 = 50%.

B only has 2 elements.

So, probability of B = 2 divided by 6, 1/3.

So, here we used the first definition - classical probability.

Now A union B.

Hopefully you remember from your middle school math class

if I have 2 sets and I build the union

I take all the elements together, so if I take A (2, 4, 6)

with the union of 1 and 5 I get 1, 2, 4, 5, 6.

Probability now of A union B of either A happening

or B is 5 divided by 6, 5 elements

divided by 6.

Notice that those 2 elements have no elements in common.

There's no number that's in A and in B.

So, they are disjoint, the intersection is the empty set.

And therefore now I can use my probability rule.

The probability of A union B is a sum of P of A and P of B

3/6 plus 2/6 is 5/6 and, guess what?

That's exactly the right answer that we saw before.

Those are now the... I showed you now the 3 axioms

of fundamental rules.

From those we can derive further rules,

some additional rules

that are very very helpful.

First, the complement rule.

What's the compliment of a set?

The compliment of an event A or the set A are all the elements in S

that are not in A.

And not surprisingly the complement rule says

the probability that the opposite of A happens is just 1 minus

the probability that A happens.

And then we have addition rules.

The general addition rule that always holds

even when A and B are not disjoined

when there's something in the intersection.

And then the rule gets a little more complicated

then the probability of A union B

if P of A plus P of B

but then now I need to subtract the probability of the intersection

because otherwise there will be some double counting.

Let's look again at our little example.

What's the opposite of an even number?

2, 4, 6, you see.

Odd numbers, so compliment of the odd numbers.

1, 3 and 5.

So probability of an odd number is 1 minus the probability

of the even numbers, 1 minus a half, "Bingo!" is a half again.

Not let's look at an event C

that has elements 1, 2, 3, 4.

A union C, now is 1, 2, 3, 4, 6

5 numbers, 5 our of 6

so the probability should get 5, 6.

Now if I use rule, probability of A union B equals P of A plus P of C

and I add A is 3/6, C has a probability of 4/6

I get 7/6.

Oh! That's not so correct probability because I'm double counting

the number 2 and 4 which are both in A and in C.

And therefore I need to subtract them out, that why we now have

this general rule, and bingo, I get again the right answer

5 divided by 6.

To summarize this lecture

I gave you formal definitions of the 3 probability concepts

that we have.

Very important, familiarize yourself of them.

It's not always a classical probability.

That we learn as kids as soon as we play a dice game

or we play a card game.

There are more important definitions

for real world decision making.

The empirical probability definition and the subjective probability

definition.

I showed you the fundamental rules, also called the axioms of probability

and finally 2 derived rules which are very helpful

in application and we will see them in action in the next couple lectures.

Thanks for your attention.

I look forward to see you back in the next lecture.