Probability¶
Probability is calculated and simulated using Python code, typically with libraries like random, numpy, or scipy. Jupyter itself is just the interactive notebook where you run these calculations.
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1. Probability Concept
Probability measures how likely an event is to occur.
Formula:
𝑃(𝐸)=
Number of favorable outcomes
Total number of outcomes
P(E)=Total number of outcomes
Number of favorable outcomes
Example: The probability of rolling a 3 on a standard die is:
𝑃(3)=16
P(3)=61
2. Calculating Probability in Python (Jupyter)
You can use Python code in Jupyter to calculate probability. For example:
# Import libraries
import random
# Simulate rolling a die 1000 times
results = [random.randint(1, 6) for _ in range(1000)]
# Probability of rolling a 3
prob_3 = results.count(3) / 1000
print("Estimated probability of rolling a 3:", prob_3)
3. Using numpy or scipy
For more advanced probability calculations:
import numpy as np
from scipy.stats import binom
# Probability of getting exactly 3 heads in 5 coin flips
prob = binom.pmf(3, n=5, p=0.5)
print("Probability of 3 heads:", prob)
binom.pmf(k, n, p) → Probability mass function for binomial events.
Cell In[1], line 12 Example: The probability of rolling a 3 on a standard die is: ^ SyntaxError: invalid non-printable character U+200B
Sample¶
Coin Tosses simulate¶
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#### import random
# Simulate 1000 coin tosses
tosses = [random.choice(['Heads', 'Tails']) for _ in range(1000)]
# Calculate probabilities
prob_heads = tosses.count('Heads') / 1000
prob_tails = tosses.count('Tails') / 1000
print("Estimated Probability of Heads:", prob_heads)
print("Estimated Probability of Tails:", prob_tails)
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import random
# Simulate 1000 coin tosses
tosses = [random.choice(['Heads', 'Tails']) for _ in range(1000)]
# Calculate probabilities
prob_heads = tosses.count('Heads') / 1000
prob_tails = tosses.count('Tails') / 1000
print("Estimated Probability of Heads:", prob_heads)
print("Estimated Probability of Tails:", prob_tails)
Estimated Probability of Heads: 0.524 Estimated Probability of Tails: 0.476
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Dice Rolls simulation¶
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# Simulate rolling a 6-sided die 1000 times
dice_rolls = [random.randint(1, 6) for _ in range(1000)]
# Probability of rolling a 3
prob_3 = dice_rolls.count(3) / 1000
print("Estimated Probability of rolling a 3:", prob_3)
Estimated Probability of rolling a 3: 0.172
Visualizing Dice Probabilities¶
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import matplotlib.pyplot as plt
# Count occurrences of each number
counts = [dice_rolls.count(i) for i in range(1,7)]
# Plot bar chart
plt.bar(range(1,7), counts)
plt.xlabel('Dice Number')
plt.ylabel('Frequency')
plt.title('Dice Roll Simulation')
plt.show()
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Binomial Distribution¶
- The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes:
Success (with probability p) Failure (with probability 1−p)
It is defined by two parameters: n = number of trials p = probability of success in a single trial
Probability Mass Function (PMF)
The probability of getting exactly 𝑘k successes in 𝑛 n trials is:
Probability¶
Explaination of the Binomialprobability formula in deatils. I want to know each formula in detal
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import itertools
import matplotlib.pyplot as plt
# Parameters
n = 3 # number of trials
p = 0.5 # probability of success
k = 2 # number of successes
# Step 1: Generate all sequences of successes (H) and failures (T)
trials = ['H', 'T']
sequences = list(itertools.product(trials, repeat=n))
# Step 2: Count sequences with exactly k successes
valid_sequences = [seq for seq in sequences if seq.count('H') == k]
# Step 3: Calculate probability
prob = len(valid_sequences) * (p**k) * ((1-p)**(n-k))
# Step 4: Display results
print(f"All possible sequences ({len(sequences)} total):")
for seq in sequences:
print(seq)
print("\nSequences with exactly 2 successes:")
for seq in valid_sequences:
print(seq)
print(f"\nP(X={k}) = {prob}")
# Step 5: Visualize sequences
colors = ['green' if seq.count('H')==k else 'lightgray' for seq in sequences]
plt.figure(figsize=(8,4))
plt.bar(range(len(sequences)), [1]*len(sequences), color=colors)
plt.xticks(range(len(sequences)), [''.join(seq) for seq in sequences])
plt.ylabel("Probability weight (conceptual)")
plt.title(f"Binomial Distribution Visualization (n={n}, k={k})\ngreen = sequences with exactly {k} successes")
plt.show()
All possible sequences (8 total):
('H', 'H', 'H')
('H', 'H', 'T')
('H', 'T', 'H')
('H', 'T', 'T')
('T', 'H', 'H')
('T', 'H', 'T')
('T', 'T', 'H')
('T', 'T', 'T')
Sequences with exactly 2 successes:
('H', 'H', 'T')
('H', 'T', 'H')
('T', 'H', 'H')
P(X=2) = 0.375
Explanation of Visualization Each bar represents one possible sequence of trials. Green bars = sequences with exactly k successes. Probability formula: P(X=k) = \underbrace{\text{# of green sequences}}_{\binom{n}{k}} \cdot p^k \cdot (1-p)^{n-k}
This makes it very clear how the formula combines “arrangements × success probability × failure probability”.
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from IPython.display import display, Math
display(Math(r'P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes} }'))
display(Math(r'P(E^c) = 1 - P(E)'))
display(Math(r'P(A \cup B) = P(A) + P(B) - P(A \cap B)'))
display(Math(r'P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}'))
$\displaystyle P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes} }$
$\displaystyle P(E^c) = 1 - P(E)$
$\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$\displaystyle P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
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import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom, norm
# Binomial Distribution
n, p = 10, 0.5
x = np.arange(0, n+1)
prob = binom.pmf(x, n, p)
plt.bar(x, prob, color='skyblue')
plt.title("Binomial Distribution (n=10, p=0.5)")
plt.show()
# Normal Distribution
mu, sigma = 0, 1
x = np.linspace(-4, 4, 1000)
plt.plot(x, norm.pdf(x, mu, sigma), color='red')
plt.title("Normal Distribution (μ=0, σ=1)")
plt.show()
Students mark¶
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import numpy as np
# Example marks of 20 students
marks = [55, 70, 80, 90, 65, 75, 85, 60, 50, 95, 88, 72, 78, 82, 68, 90, 77, 66, 84, 91]
Frequency Distribution¶
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# Define bins (intervals) for marks
bins = [0, 50, 60, 70, 80, 90, 100]
# Count number of students in each bin
freq, bin_edges = np.histogram(marks, bins=bins)
print("Frequency:", freq)
print("Bins:", bin_edges)
Frequency: [0 2 4 5 5 4] Bins: [ 0 50 60 70 80 90 100]
Calculate probabilities¶
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probabilities = freq / len(marks)
print("Probabilities:", probabilities)
Probabilities: [0. 0.1 0.2 0.25 0.25 0.2 ]
Visualize probability distribution¶
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import matplotlib.pyplot as plt
# Midpoint of each bin for plotting
mid_points = (bin_edges[:-1] + bin_edges[1:]) / 2
plt.bar(mid_points, probabilities, width=8, color='skyblue', edgecolor='black')
plt.xlabel("Marks Range")
plt.ylabel("Probability")
plt.title("Probability Distribution of Students' Marks")
plt.show()
Cumulative Probability¶
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cumulative_prob = np.cumsum(probabilities)
print("Cumulative Probabilities:", cumulative_prob)
plt.plot(mid_points, cumulative_prob, marker='o', color='red')
plt.xlabel("Marks Range")
plt.ylabel("Cumulative Probability")
plt.title("Cumulative Probability of Students' Marks")
plt.grid(True)
plt.show()
Cumulative Probabilities: [0. 0.1 0.3 0.55 0.8 1. ]
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✅ Explanation
Histogram bins → divide marks into intervals (like 50–60, 60–70…).
Frequency → number of students in each bin.
Probability → frequency divided by total students.
Visualization: bar chart for probability, line chart for cumulative probability
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