1.6. Two Special Events

Last updated: January 13th, 20202020-01-13Project preview
In [1]:
import pandas as pd

Two Special Events

There are two particularly important events that we must notice:

The event of "nothing happening", which means we make an experiment and nothing happens. We are going to represent it with the empty set: $\emptyset$.

The event of "anything happening", which means we make an experiment and something happens. We are going to represent it with the full set Omega: $\Omega$.

Of course that it is impossible that nothing happens when we perform an experiment: we might not know what will happen if we are performing a random experiment, but we certainly know that something will.

We have already said that the probability of impossible outcomes is $0$ (remember the probability scale) so, as expected, the probability of $\emptyset$ will be $0$.

On the other hand, when we perform an experiment, we are certain that we will get some outcome, something will happen. So if we just want anything to happen, we can be sure that will be the case.

And we have said that the probability of certain outcomes is $1$ (remember the probability scale) so, as expected the probability of $\Omega$ will be $1$.

Summarising:

$P(\emptyset)=0$ and $P(\Omega)=1$.

Before going to practise, let's see this in our example:

In [2]:
dataset =  pd.DataFrame({
    'Person #':[1,2,3,4,5,6,7,8,9,10],
    'City':['SF','SF','NY','NY','NY','SF','NY','SF','SF','SF'],
    'Age':[41,26,28,53,32,51,65,49,25,33]
})
dataset
Out[2]:
Person # City Age
0 1 SF 41
1 2 SF 26
2 3 NY 28
3 4 NY 53
4 5 NY 32
5 6 SF 51
6 7 NY 65
7 8 SF 49
8 9 SF 25
9 10 SF 33
In [3]:
dataset.sample(1)['Age'].values[0]
Out[3]:
26

Now imagine we have the event $E=$ "the person chosen is younger than $20$". We can tell that this event actually contains no age at all, because our ages start at $25$. So $P(E)=P(\emptyset)=0$.

In the case of, for example, the event $F=$ "the person chosen is younger than $70$" we can see that it contains all possible ages, because our ages finish at $65$. So $P(F)=P(\Omega)=1$.

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