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Probability Basics 1


Probability Basics 1

The Probability of an event means the likelihood that the specific event will take place.
In English,we use the words probable and improbable fairly loosely - saying something is probable just means that it is more likely to occur than not, while saying that it is improbable means that is more likely to not happen than to happen.
However, in maths, we prefer to be more precise in our definition and hence we need to establish a sound mathematical foundation for the concept of Probability.

Let us try to establish such a foundation:
Suppose a normal coin is tossed, and while it is in the air, you call out "Heads".
What are the chances that you will win the toss?
Ignoring unrealistic results like the coin landing on its side, there are only two things that can happen — it can either show Heads or Tails. And in one case (Heads) you win, while in the other (Tails) you lose.
We can hence say that, in 1 out of 2 cases, you will win. "The probability of winning is 50 %”
(Note: the above is true only in the case of a "fair" or "unbiased" coin. If you were to use the coin of Harvey Dent in the Dark Knight, or that of Jai in Sholay, the answer would be very different! )

Let's consider another example:
Suppose an unbiased cubic die with 6 faces (numbered l, 2, 3, 4, 5, 6) is rolled and you bet on the number 4.
The roll of a die has 6 possible outcomes, and in only 1 of them will you win the bet. Hence we can say that the Probability of winning is 1/6.

In both the above cases, to find the probability, we took the total number of things that could happen, and we saw what fraction (or percentage) of them satisfied the required conditions.

Now let us try to couch this in more formal terminology, through the following definitions:

Sample Space (S):
Whenever any action is taken which has multiple possible outcomes, the set of all possible outcomes is described as the Sample Space.
In the above example of a coin toss, the sample space could be defined as S — {H, T}. In the case of the die being rolled, we could define a sample space of S = (l, 2, 3, 4, 5, 6) or a sample space of S = {even,odd}.

This is the set of "desired results" of the action.
For example if a die is rolled and you bet on "odd numbers", then the desired Event or set of favourable outcomes is E = (1, 3, 5). (Note: a favourable event just means "the event whose probability is sought" - it might be quite unfavourable for the protagonist of the question! For example if we want to know "what is the probability that someone will die in the next episode of some web series?" the favourable outcome is "someone dying" even though that is very evidently not favourable for the individual character in question!)

Given the above two definitions, we can now proceed to define the probability of an event as follows:

The Probability Of an event E can be computed as:
Number of Favourable Outcomes / Total number of outcomes
i.e. P(E) = n(E)/n(S)
[where n(A) is the number of elements in set A]

Now that we have a basic definition in place, we need to understand how to apply it.
In particular, it is very essential to define the Sample Space properly.

Let's understand this through an example:
If a fair die is tossed, what is the probability that it will show
(a) a prime number?
(b) a composite number?

Let's now list out some possible sample spaces and figure out whether they would be valid or not:
If we take the standard sample space (l, 2, 3, 4, 5, 6) we will get the primes 2, 3 and 5 and the composites 4 and 6.
So the probability of getting a prime is 3/6 or 1/2 while that for a composite is 2/6 or 1/3 .

What if we take the sample space as {Prime, Composite}?

This would give us P(Prime) = 1/2 but P(Composite) = 1/2 as well.
This doesn't match our earlier answer; something must be wrong!
One problem we can immediately Recognize is that this sample space does not cover everything!
The number 1 is not included in it.
So it is not a valid sample space.

Let's correct it:

What if we take the sample space as {Prime, Composite, Odd}?
This covers everything, and gives us P(Composite) = 1/3 ; however, it gives us P(Prime) = 1/3 as well.
This also seems problematic.
Looking at this sample space, we see that some numbers occur more than once. For example 3 and 5 occur both in Prime and in Odd. Again, this is not a valid Sample Space.
What if we take the sample space as {Prime, Composite, l}? This covers everything, and that too exactly once each. But again it gives us P(Composite) = 1/3 and P(Prime) 1/3 as well.
(And it gives us P(l) = 1/3 instead of the 1/6 we ought to expect. So something is still not matching.
Here, the problem is that there is only one way to get a l, two ways to get a composite, and three ways to get a prime. A prime is thrice as likely to occur as a l. So a third requirement is that the elements of a Sample Space should also be equally likely to happen.

Thus, the elements Of a valid sample space must satisfy the following three criteria:

(a) Exhaustive: Every possible outcome has to be included
(b) Mutually Exclusive: No outcome should occur more than once (i.e. there should be no overlap)
(c) Equiprobable: All the elements should be equally likely to occur

For example, in the case of the die above, the first possibility we considered {l, 2, 3, 4, 5, 6} is a valid Sample Space, satisfying all the three conditions. {Even, Odd} is another such valid Sample Space.

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