Data ScienceIntermediate16 min readUpdated March 2025

Probability & Distributions

Probability is the language of uncertainty. Learn core rules, conditional probability and Bayes theorem, then meet the distributions — normal, binomial and Poisson — that model real data.

Probability Basics

The probability of an event is a number between 0 and 1. Key rules:

Complement — P(not A) = 1 - P(A).
Addition — P(A or B) = P(A) + P(B) - P(A and B).
Independence — P(A and B) = P(A) x P(B) when events do not affect each other.
Conditional — P(A | B) = P(A and B) / P(B).

Bayes' Theorem

Bayes' theorem updates a belief given new evidence: P(A | B) = P(B | A) x P(A) / P(B).

It explains why a positive result on a very accurate test can still be more likely to be a false alarm when the condition is rare.

python

# Disease affects 1% of people. Test: 95% sensitivity, 90% specificity.
p_disease = 0.01
sensitivity = 0.95            # P(+ | disease)
false_positive = 0.10         # P(+ | no disease)

p_positive = sensitivity * p_disease + false_positive * (1 - p_disease)
p_disease_given_positive = sensitivity * p_disease / p_positive

print(f"P(disease | positive) = {p_disease_given_positive:.1%}")  # ~8.8%

Common Distributions

Distributions describe how values are spread out:

Normal — Bell curve, defined by mean and standard deviation; heights, measurement error.
Binomial — Number of successes in n independent yes/no trials.
Poisson — Count of events in a fixed interval, e.g. support tickets per hour.
Uniform — Every value in a range equally likely.

Working with Distributions in SciPy

SciPy provides the PDF/PMF, CDF and random sampling for each distribution:

python

from scipy import stats

# Normal: P(score between 70 and 90) with mean 75, sd 10
norm = stats.norm(loc=75, scale=10)
print(norm.cdf(90) - norm.cdf(70))          # ~0.62

# Binomial: P(exactly 7 conversions out of 20 visits at 25%)
print(stats.binom.pmf(7, n=20, p=0.25))     # ~0.11

# Poisson: P(more than 5 tickets in an hour, avg 3)
print(1 - stats.poisson.cdf(5, mu=3))       # ~0.084

samples = norm.rvs(size=1000, random_state=0)
print(samples.mean(), samples.std())

Key Takeaways

Conditional probability and Bayes theorem update beliefs with evidence.
Base rates matter — rare events produce many false positives.
Choose a distribution that matches how the data is generated.
SciPy gives you PDFs, CDFs and sampling for every distribution.

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