Nipun Batra
February 2, 2025
probability, sample space, events, combinatorics, probability law, event space
This notebook covers the fundamental concepts of probability theory, including sample spaces, events, and probability laws. We’ll explore these concepts through practical examples using Python.
Let’s begin by exploring sample spaces for common probability experiments:
import numpy as np
# Define the sample space and event space for a fair coin
sample_space = np.array(['H', 'T'])
print("Sample Space for Coin Toss:", sample_space)
# Dice throw
sample_space_dice = [1, 2, 3, 4, 5, 6]
print("Sample Space for Dice Throw:", sample_space_dice)Sample Space for Coin Toss: ['H' 'T']
Sample Space for Dice Throw: [1, 2, 3, 4, 5, 6]In probability theory, the sample space (Ω) is the set of all possible outcomes of an experiment. An event is any subset of the sample space.
Let’s start with simple examples:
Next, let’s define some specific events:
To understand all possible events, we need to explore combinatorics. The event space (σ-algebra) consists of all possible events, which is the power set of the sample space.
Let’s explore combinations and permutations using Python’s itertools:
# Mini tutorial on itertools
from itertools import combinations, permutations, product
x = [1, 2, 3, 4]
# Combinations
print('Combinations')
print(list(combinations(x, 2)))Combinations
[(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]for i in range(0, len(x)+1):
print("Combinations of length ", i)
print(list(combinations(x, i)))
print()Combinations of length 0
[()]
Combinations of length 1
[(1,), (2,), (3,), (4,)]
Combinations of length 2
[(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)]
Combinations of length 3
[(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)]
Combinations of length 4
[(1, 2, 3, 4)]
# Combine all using chain
from itertools import chain
powerset = list(chain.from_iterable(combinations(x, i) for i in range(0, len(x)+1)))
print(powerset)[(), (1,), (2,), (3,), (4,), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2, 3, 4)]import itertools
# Generate the entire event space (power set)
def generate_event_space(sample_space):
"""
Generates the power set of the sample space, which is the event space.
Args:
sample_space (np.ndarray): The sample space.
Returns:
list: A list of NumPy arrays representing all possible events.
"""
n = len(sample_space)
# Use itertools to generate all subsets
power_set = list(itertools.chain.from_iterable(
itertools.combinations(sample_space, r) for r in range(n + 1)
))
# Convert tuples to NumPy arrays
return [np.array(event) for event in power_set][array([], dtype=float64),
array(['H'], dtype='<U1'),
array(['T'], dtype='<U1'),
array(['H', 'T'], dtype='<U1')][array([], dtype=float64),
array([1]),
array([2]),
array([3]),
array([4]),
array([5]),
array([6]),
array([1, 2]),
array([1, 3]),
array([1, 4]),
array([1, 5]),
array([1, 6]),
array([2, 3]),
array([2, 4]),
array([2, 5]),
array([2, 6]),
array([3, 4]),
array([3, 5]),
array([3, 6]),
array([4, 5]),
array([4, 6]),
array([5, 6]),
array([1, 2, 3]),
array([1, 2, 4]),
array([1, 2, 5]),
array([1, 2, 6]),
array([1, 3, 4]),
array([1, 3, 5]),
array([1, 3, 6]),
array([1, 4, 5]),
array([1, 4, 6]),
array([1, 5, 6]),
array([2, 3, 4]),
array([2, 3, 5]),
array([2, 3, 6]),
array([2, 4, 5]),
array([2, 4, 6]),
array([2, 5, 6]),
array([3, 4, 5]),
array([3, 4, 6]),
array([3, 5, 6]),
array([4, 5, 6]),
array([1, 2, 3, 4]),
array([1, 2, 3, 5]),
array([1, 2, 3, 6]),
array([1, 2, 4, 5]),
array([1, 2, 4, 6]),
array([1, 2, 5, 6]),
array([1, 3, 4, 5]),
array([1, 3, 4, 6]),
array([1, 3, 5, 6]),
array([1, 4, 5, 6]),
array([2, 3, 4, 5]),
array([2, 3, 4, 6]),
array([2, 3, 5, 6]),
array([2, 4, 5, 6]),
array([3, 4, 5, 6]),
array([1, 2, 3, 4, 5]),
array([1, 2, 3, 4, 6]),
array([1, 2, 3, 5, 6]),
array([1, 2, 4, 5, 6]),
array([1, 3, 4, 5, 6]),
array([2, 3, 4, 5, 6]),
array([1, 2, 3, 4, 5, 6])]# Probability law function for a fair coin
def probability(event, sample_space):
"""
Computes the probability of an event for a fair coin.
Args:
event (np.ndarray): The event (subset of the sample space).
Returns:
float: The probability of the event.
"""
# Convert the event into a NumPy array for comparison
event = np.array(event)
# Validate if the event is a subset of the sample space
if not np.all(np.isin(event, sample_space)):
raise ValueError("Invalid event. Event must be a subset of the sample space.")
# Probability logic
if len(event) == 0: # Empty set
return 0.0
elif np.array_equal(event, sample_space): # Entire sample space
return 1.0
else: # Any single event (like {H} or {T})
return len(event) / len(sample_space)for event in generate_event_space(sample_space):
print(f"Event: {event} -> Probability: {probability(event, sample_space)}")
Event: [] -> Probability: 0.0
Event: ['H'] -> Probability: 0.5
Event: ['T'] -> Probability: 0.5
Event: ['H' 'T'] -> Probability: 1.0import pandas as pd
# Initialize an empty list to store events and probabilities
events_data = []
# Generate events and probabilities
for event in generate_event_space(sample_space_dice):
event_tuple = tuple(event)
event_prob = probability(event, sample_space_dice)
events_data.append((event_tuple, event_prob))
# Create a pandas DataFrame
df = pd.DataFrame(events_data, columns=['Event', 'Probability'])