Probability Essentials — FAANG-Level Lab

Goal: ML-relevant probability: expectation, variance, Bayes, and simulation checks.

Outcome: You can reason about uncertainty, distributions, and Bayes updates (interview-ready).

Section 1 — Discrete Random Variables

Task 1.1: Expectation & variance from a PMF

Given values x and probabilities p (sum to 1):

• ●

  implement E[X] and Var(X)

• ●

  E[X] = sum p_i x_i

• ●

  Var(X) = E[X^2] - (E[X])^2

Explain: Why is variance not linear, but expectation is?

Section 2 — Conditional Probability & Bayes

Task 2.1: Bayes theorem (classic interview)

Disease test example:

• ●prevalence P(D)=0.01
• ●sensitivity P(+|D)=0.99
• ●false positive rate P(+|~D)=0.05 Compute P(D|+)

P(D|+) = P(+|D)P(D) / (P(+|D)P(D) + P(+|~D)P(~D))

FAANG gotcha: base-rate fallacy.

Task 2.2: Simulation check (sanity)

Simulate N people and estimate P(D|+) empirically.

• ●sample disease ~ Bernoulli(P_D)
• ●sample test result conditional on disease

Explain: Why does simulation converge to the analytic value?

Section 3 — Continuous Distributions (Normal)

Task 3.1: Standardization (z-score)

Given X ~ Normal(mu, sigma^2). Compute standardized Z=(X-mu)/sigma.

• ●simulate X and check Z mean ~0, std ~1

ML link: standardization shows up in preprocessing and SGD stability.

Section 4 — Naive Bayes Thinking (Optional Mini)

Task 4.1: Compute log-odds for a toy Naive Bayes

Given word likelihoods for spam vs ham, compute posterior odds for a message.

• ●work in log space (sum logs)

FAANG gotcha: multiplying small probabilities underflows; use logs.