> ## Documentation Index
> Fetch the complete documentation index at: https://ai.tharung.in/llms.txt
> Use this file to discover all available pages before exploring further.

# Probability example

1. Basic Introduction<br /><br />Probability is the mathematics of uncertainty.<br /><br />AI/ML uses probability to:<br />- predict outcomes<br />- handle uncertain data<br />- build models

```text theme={null}
# Probability of getting head in coin toss

probability = 1 / 2
print(probability)
```

***

# 2. Basic Terminology

| Term         | Meaning                      |
| ------------ | ---------------------------- |
| Experiment   | Action with uncertain result |
| Sample Space | All possible outcomes        |
| Event        | Selected outcomes            |
| Probability  | Chance of occurrence         |

Example:

```python theme={null}
outcomes = ["Head", "Tail"]

print(len(outcomes))
```

***

# 3. Types of Events

## Simple Event

Single outcome.

Example:

```text theme={null}
Rolling dice = 6
```

## Compound Event

Multiple outcomes.

Example:

```text theme={null}
Rolling dice = even number
(2,4,6)
```

```python theme={null}
event = [2,4,6]
print(event)
```

***

# 4. Probability Rules

## Formula

```text theme={null}
P(A) = Favorable Outcomes / Total Outcomes
```

Example:

```python theme={null}
favorable = 3
total = 6

p = favorable / total

print(p)
```

Output:

```text theme={null}
0.5
```

***

# 5. Conditional Probability

Probability of A happening when B is true.

Formula:

```text theme={null}
P(A|B)=P(A∩B)/P(B)
```

Example:

```python theme={null}
A_and_B = 2
B = 5

conditional_probability = A_and_B / B

print(conditional_probability)
```

***

# 6. Law of Total Probability

Formula:

```text theme={null}
P(A)=P(A|B)P(B)+P(A|C)P(C)
```

Example:

```python theme={null}
p_a_b = 0.5
p_b = 0.4

p_a_c = 0.2
p_c = 0.6

result = (p_a_b*p_b)+(p_a_c*p_c)

print(result)
```

***

# 7. Bayes' Theorem

Used heavily in ML classification.

Formula:

```text theme={null}
P(A|B)=P(B|A)P(A)/P(B)
```

Example:

```python theme={null}
p_b_a = 0.9
p_a = 0.1
p_b = 0.2

bayes = (p_b_a*p_a)/p_b

print(bayes)
```

***

# 8. Random Variable

A variable whose value depends on random outcomes.

Types:

* Discrete
* Continuous

Example:

```python theme={null}
import random

x = random.randint(1,6)

print(x)
```

***

# 9. Mean, Median, Mode

## Mean

Average value.

```text theme={null}
Mean = Sum / Count
```

Example:

```python theme={null}
data=[2,4,6]

mean=sum(data)/len(data)

print(mean)
```

***

## Median

Middle value.

```python theme={null}
import statistics

data=[1,3,5]

print(statistics.median(data))
```

***

## Mode

Most frequent value.

```python theme={null}
import statistics

data=[1,2,2,3]

print(statistics.mode(data))
```

***

# 10. Variance & Standard Deviation

Variance measures spread.

Standard deviation is square root of variance.

Example:

```python theme={null}
import statistics

data=[10,20,30]

variance=statistics.variance(data)

std=statistics.stdev(data)

print(variance)
print(std)
```

***

# 11. Probability Distributions

A probability distribution describes possible values and their probabilities.

Types:

* Bernoulli
* Binomial
* Uniform
* Normal
* Poisson

***

# 12. Bernoulli Distribution

Only two outcomes:

```text theme={null}
Success = 1
Failure = 0
```

Example:

```python theme={null}
from scipy.stats import bernoulli

x = bernoulli.rvs(0.5)

print(x)
```

***

# 13. Binomial Distribution

Number of successes in repeated trials.

Formula:

```text theme={null}
P(X)=nCx p^x(1-p)^(n-x)
```

Example:

```python theme={null}
from scipy.stats import binom

result = binom.pmf(
    k=3,
    n=5,
    p=0.5
)

print(result)
```

***

# 14. Uniform Distribution

All values have equal probability.

Example:

```python theme={null}
from scipy.stats import uniform

x = uniform.rvs(
    loc=0,
    scale=10
)

print(x)
```

***

# 15. Normal Distribution

Most common distribution in ML.

Shape:

```text theme={null}
       *
     *   *
   *       *
 *           *
--------------
```

Example:

```python theme={null}
from scipy.stats import norm

x = norm.pdf(
    0,
    loc=0,
    scale=1
)

print(x)
```

***

# 16. Poisson Distribution

Used for counting events.

Examples:

* website visits
* failures
* calls per hour

Formula:

```text theme={null}
P(X)= (λ^x e^-λ)/x!
```

Example:

```python theme={null}
from scipy.stats import poisson

result = poisson.pmf(
    k=3,
    mu=5
)

print(result)
```

***

# AI/ML Probability Summary

| Concept                 | ML Usage         |
| ----------------------- | ---------------- |
| Probability             | Prediction       |
| Conditional Probability | Classification   |
| Bayes Theorem           | Naive Bayes      |
| Random Variable         | Data Modeling    |
| Mean                    | Feature Analysis |
| Variance                | Feature Scaling  |
| Normal Distribution     | Statistics       |
| Binomial                | Binary Models    |
| Poisson                 | Event Prediction |