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https://github.com/sanju-srivatsa/probability-and-statistics


https://github.com/sanju-srivatsa/probability-and-statistics

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README 

        
# Probability-and-Statistics



---



### **Detailed Notes on Probability and Statistics**



#### **1. Introduction to Probability**



**What is Probability?**



Probability is a measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1:

- **0** indicates the event will not happen.

- **1** indicates the event will definitely happen.



**Probability Formula:**

\[

P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}

\] 



**Example:** If you flip a coin, the probability of getting heads is:

\[

P(\text{Heads}) = \frac{1}{2} = 0.5

\]



**Sample Space (S):**

The sample space is the set of all possible outcomes of an experiment.



- S or Ω might be used to denote the sample space.



**Example:** For a die roll, the sample space is \( S = \{1, 2, 3, 4, 5, 6\} \).



#### **2. Types of Probability**



**1. Classical Probability:**

- Based on the assumption that all outcomes are equally likely.

- **Example:** The probability of drawing an ace from a standard deck of 52 cards is:

\[

P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}

\]



**2. Empirical Probability:**

- Based on observed data from experiments or historical records.

- **Example:** If it rains on 30 out of 100 days, the empirical probability of rain is:

\[

P(\text{Rain}) = \frac{30}{100} = 0.3

\]



**3. Subjective Probability:**

- Based on personal judgment or experience rather than data.

- **Example:** A doctor might say, "There’s a 90% chance that this treatment will work," based on their experience.



#### **3. Definitions - Sets and Elements**



A **Set** is a collection of distinct elements or members, often representing possible outcomes.



**Notation:**

- A set is denoted by a capital letter, e.g., \( A = \{a, b, c, d\} \).

- An element belonging to a set is written as \( a \in A \).



**Defining a Set:**

1. **Listing all the elements:** \( A = \{a, b, c, d\} \)

2. **Describing the properties:** e.g., "A is the set of all vowels in the English alphabet."



#### **4. Definitions - Sample Spaces and Factorial**



**Sample Space (S):**

- A set of all possible outcomes of an experiment.

- **Example:** For a coin toss, \( S = \{\text{HEADS}, \text{TAILS}\} \).



**Factorial (n!):**

- The product of all positive integers up to \( n \).

- **Example:** For three colored marbles, the number of ways they can be arranged is \( 3! = 3 \times 2 \times 1 = 6 \).



#### **5. Definitions - Events**



An **Event** is a subset of a sample space that includes one or more outcomes.



**Example:** 

- If you roll a die, \( S = \{1, 2, 3, 4, 5, 6\} \).

- An event, such as rolling an even number, is \( E = \{2, 4, 6\} \).



**Probability of an Event:**

\[

P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in the sample space}}

\]

**Example:** The probability of rolling an even number on a die is:

\[

P(\text{Even}) = \frac{3}{6} = \frac{1}{2}

\]



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