Till now, we have studied about statistics and different tools we use to study about it. We recall that** Statistics** is the science of looking at a given data and understanding the story of the data. On the other hand, **Probability **is the study of the story and understanding the data behind it. There is a **complementary relationship existing between the two.**

We can define probability as the **language **to describe the relationship between the data and the underlying causes. ** **

Probability is the language for quantifying uncertainty. -Larry Wasserman

Before we dive into probability, it is imperative for us to understand some important terms associated with it.

## Sample Spaces

The **sample space** sigma is the set of all possible outcomes of an experiment. All such **outcomes **are known as **elements. **Any subset of the **sample space** is known as **Events.**

Ex.: If we toss a coin twice, then the sample space will be:

where H stands for heads and T for tails.

The event *A* that the first toss is heads is:

As described earlier, we can use probability to answer our question: **What is the chance that the first toss is heads?**

Out of the sample space consisting of 4 events, we have 2 events where the **first toss is a head., { HH, HT }. **Since the chance of occurrence of all these 4 outcomes is equal, we can say the probability of any one outcome is 1/4 = 0.25 and for the probability of occurrence of the event, **first toss is heads ** is 2/4 = 0.5.

We can also get another important result:

where O is standing for outcome and P(O_{i}) stands for the probability of occurrence of i^{th }outcome. The total sum of probability of occurrences of all the outcomes is 1.

## Complementary probability

## Q) If the probability of getting a head is 0.45, then what will be the probability of not getting heads, i .e. getting tails.

Ans) From the above, we can see that the total probability of all the outcomes belonging to the sample space is 1. So, the probability of not getting heads can be given as:

In **general, **P( ~H ) + P ( H ) = 1, where P (~ H ) is known as the **complementary event **to H.

# Intersection of events

## Q) What is the probability of getting at least 1 head in a coin tossed twice and getting at least 1 tail?

Ans) This event can be thought of as the case when both events, i.e. getting 1 head in a coin tossed twice and getting at least 1 tail is occurring simultaneously.

getting at least 1 head = { HH, HT, TH }

P ( getting at least 1 head ) = 0.25 + 0.25 + 0.25 = 0.75

getting at least one tail = { HT, TH, TT }

getting at least 1 head AND getting at least 1 tail = { HT, TH }

P ( getting at least 1 head & getting at least 1 tail ) = 0.25 + 0.25 = 0.5

So, P( A ^ B ) is also defined as the **intersection **of two events, i.e. probability of the outcomes satisfying both the events A and B.

# Disjoint events

We say that 2 events are disjoint when P ( AA^ B) = 0,i.e. the events A and B cannot occur simultaneously.

## Q) What is the probability of getting 2 heads and 2 tails when a coin is flipped twice?

Ans) We can define the events, of getting 2 heads and getting 2 tails separately as:

A := getting 2 heads: { HH }

B := getting 2 tails: { TT }

Clearly, there is no way that both of these events can occur simultaneously.

A ^ B := getting 2 heads and 2 tails = { }

P( A ^ B ) = 0

A and B are **mutually exclusive or disjoint events.**

## Union of events

From the set theory notation of a union of 2 events, we can say that the union of 2 events A and B can be defined as:

## Q) What is the probability of getting either 2 heads or 2 tails in flipping a coin twice?

Ans) We can write this event as the sum of the probabilities of getting 2 heads and sum of getting 2 tails.

getting 2 heads = { HH}

getting 2 tails = { TT }

getting 2 heads or 2 tails = { HH, TT }

P ( getting 2 heads or 2 tails ) = P ( getting 2 heads ) + P ( getting 2 tails ) - P ( getting 2 heads and 2 tails )

P ( getting 2 heads or 2 tails ) = 0.25 + 0.25 - 0 = 0.5

So, P ( A U B ) is also defined as the **union **of two events, or probability of either A or B occurring.

## Key Takeaways:

1) Probability is the language to describe the relationship between the data and the underlying causes.

2) The **sample space** sigma is the set of all possible outcomes of an experiment. All such **outcomes **are known as **elements. **Any subset of the **sample space** is known as **events.**

3) **Total probability** of all the outcomes of sample space is 1.

4) Probability of **Complement **of event A is the sum of all the outcomes not satisfying event A. It can be written as: P (~ A ) = 1 - P(A).

5) P ( A U B ) is also defined as the **union **of two events, or sum of the probabilities of all the outcomes satisfying either event A or event B.

6) P( A ^ B ) is also defined as the **intersection **of two events, i.e. probability of the outcomes satisfying both the events A and B.

7) 2 events A and B are called **mutually exclusive ** or **disjoint **if

P ( A ^ B) is 0.

## References:

1) Udacity

2) A Concise Course in Statistical Inference (Springer Texts in Statistics)

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