Section 02 Foundations & Computational Lab

Addition Rules & Counting Principles

Mutually exclusive events, general addition theorems, multi-category frequency tables, permutations, and combinatorial probability with Python 3.

Warm-Up Problem Solutions

Lab 01 Tasks: Model Review

TASK 01

Complement Defect Model: Defect rate p = 0.08, n = 4. Probability of at least one defect.

1 - 0.92⁴ ≈ 0.28361

TASK 02

Badge Key Permutations: 2 uppercase letters + 5 digits without repetition. P(Crack) = 4.89e-8.

26² × ₁₀P₅ = 20,442,240

TASK 03

Dependent 3-Card Draw: King then Queen then Jack without replacement.

(4/52) × (4/51) × (4/50) ≈ 0.000483

# Task 1: At least 1 defective itemp_defect = 0.08p_at_least_one = 1 - (1 - p_defect) ** 4print("Task 1:", round(p_at_least_one, 5))# Task 2: Security Badge Permutationskeys = (26 ** 2) * (10 * 9 * 8 * 7 * 6)p_crack = 1 / keysprint(f"Task 2 Keys: {keys} | P: {p_crack:.2e}")# Task 3: 3-Card Dependent Drawp_cards = (4 / 52) * (4 / 51) * (4 / 50)print("Task 3:", round(p_cards, 6))

CONSOLE OUTPUTEXIT CODE: 0

Task 1: 0.28361
Task 2 Keys: 20442240 | P: 4.89e-08
Task 3: 0.000483

Event Disjointness

Mutually Exclusive Events

ADDITION RULEP(A or B) = P(A) + P(B)

CONDITIONEvents cannot happen on the same trial (disjoint)

STEP 01

Fours and Aces: 4 fours and 4 aces in standard 52-card deck

P(4) = 4/52, P(Ace) = 4/52

STEP 02

Direct Sum: Add mutually exclusive probabilities without overlap

8/52 ≈ 0.15385

# Mutually exclusive: 4 of a kind OR Acep_four = 4 / 52p_ace = 4 / 52probability = p_four + p_aceprint("P(4 or Ace):", round(probability, 5))

CONSOLE OUTPUTEXIT CODE: 0

P(4 or Ace): 0.15385

Set Union & Overlap

General Addition Rule

UNION THEOREMP(A ∪ B) = P(A) + P(B) - P(A ∩ B)

OVERLAP NOTESubtract joint probability once to avoid double counting

STEP 01

Identify Overlap: Event A below 3 and Event B odd share outcome 1

A ∩ B = {1}

STEP 02

Apply Theorem: (2/6) + (3/6) - (1/6) = 4/6 ≈ 0.6667

2/6 + 3/6 - 1/6 = 4/6

# Roll die: A = below 3 {1,2}, B = odd {1,3,5}p_a = 2 / 6p_b = 3 / 6p_overlap = 1 / 6probability = p_a + p_b - p_overlapprint("P(A or B):", round(probability, 4))

CONSOLE OUTPUTEXIT CODE: 0

P(A or B): 0.6667

Two-Way Table Probability

Blood-Type Addition Counts

COUNT RULEP(A ∪ B) = [n(A) + n(B) - n(A ∩ B)] / n(total)

STEP 01

Substitute Counts: Type B (45) + Rh-Neg (65) - Overlap (8) = 102

45 + 65 - 8 = 102

STEP 02

Calculate Ratio: Divide favorable category sum by sample size (409)

102 / 409 ≈ 0.2494

# Blood sample: B=45, Rh-neg=65, both=8, total=409favorable = 45 + 65 - 8total = 409probability = favorable / totalprint("P(B or Rh-):", round(probability, 4))

CONSOLE OUTPUTEXIT CODE: 0

P(B or Rh-): 0.2494

Counting Principles

Factorials & Permutations

FACTORIALn! = n × (n - 1) × (n - 2) × ⋯ × 1

PERMUTATIONₙPᵣ = n! / (n - r)!

SUDOKU

Sudoku Row: 9 distinct digits in 9 positions: 9×8×7×6×5×4×3×2×1

9! = 362,880

3-DIGIT CODE

Code without repeats: 10 choices × 9 choices × 8 choices

10 × 9 × 8 = 720

import math# Sudoku 9-digit arrangementsudoku = math.factorial(9)# 3-digit code from 10 digits without repeatscodes = 10 * 9 * 8print("Sudoku:", sudoku)print("Codes:", codes)

CONSOLE OUTPUTEXIT CODE: 0

Sudoku: 362880 Codes: 720

Combinatorics

Combinations Formula

COMBINATIONC(n, r) = n! / [r! (n - r)!]

MULTINOMIAL12 houses (6, 4, 2): 12! / (6! 4! 2!) = 13,860

SELECTION

Choose 4 Companies from 16: (16×15×14×13)/(4×3×2×1) = 1,820

C(16, 4) = 1,820

import math# Choose 4 from 16 companiesgroups = math.comb(16, 4)print("4-Company Selections:", groups)# Multinomial house arrangementshouses = math.factorial(12) // ( math.factorial(6) * math.factorial(4) * math.factorial(2))print("House Arrangements:", houses)

CONSOLE OUTPUTEXIT CODE: 0

4-Company Selections: 1820 House Arrangements: 13860

Sample Selection Model

Combination Probability

PROBABILITYP = C(3, 1) × C(397, 3) / C(400, 4)

FAVORABLE

Affected & Unaffected: C(3,1) × C(397,3)

3 × 10,349,790 = 31,049,370

TOTAL & RATIO

All Selections: C(400,4) = 1,050,739,900; Ratio = 0.02955

P ≈ 0.02955

import mathfavorable = math.comb(3, 1) * math.comb(397, 3)total = math.comb(400, 4)probability = favorable / totalprint("Probability:", round(probability, 5))

CONSOLE OUTPUTEXIT CODE: 0

Probability: 0.02955