Probability Foundations Activity Packet
Finite Mathematics • Unit 3: Probability
Probability Foundations Activity Packet
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Date: Period:
Day 1 Simple Events, Complements, & The Addition Rule
Materials: Standard 6-Sided Die • Standard Deck of Cards
Theoretical Probability
Equally likely outcomes in sample space \(S\):
\(P(E) = \frac{n(E)}{n(S)} = \frac{\text{favorable}}{\text{total}}\)
Complement Rule
Probability of event \(E\) NOT happening:
\(P(E') = 1 - P(E)\)
Addition Rule
Union of two events (\(A \cup B\)):
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
1 Lab Investigation: The Single Die Sample Space
Sample Space \(S = \{1, 2, 3, 4, 5, 6\}\)
Roll a single 6-sided die 12 times. Tally your actual outcomes in the table below, then calculate both the experimental and theoretical probabilities.
| Face Value | 1 | 2 | 3 | 4 | 5 | 6 | Total Rolls |
|---|
| Tally / Frequency | | | | | | | 12 |
a) Theoretical \(P(\text{Prime Number})\) • Note: \(\{2, 3, 5\}\):
b) Experimental probability of rolling a Prime from your 12 rolls:
c) Complement: Theoretical \(P(\text{NOT Prime})\) using \(1 - P(\text{Prime})\):
d) Explain why \(P(\text{Prime}) + P(\text{NOT Prime}) = 1\):
2 Playing Card Investigation: Mutually Exclusive vs. Overlapping
Standard 52-Card Deck
In a single random draw of one card from a well-shuffled 52-card deck, examine whether the two events can happen at the same time (\(A \cap B \neq \emptyset\)) or are mutually exclusive (\(A \cap B = \emptyset\)).
Case A: Drawing a King (\(K\)) OR drawing a Queen (\(Q\)) Mutually Exclusive
Can a single card be both a King and a Queen? Explain, then calculate \(P(K \cup Q)\):
Case B: Drawing a King (\(K\)) OR drawing a Heart (\(H\)) Overlapping
Identify the intersection \(K \cap H\). Why would adding \(P(K) + P(H)\) result in double-counting?
Finite Mathematics • Student Investigation Packet Page 1 of 6
Day 1 Continued
Addition Rule Models: Venn Diagrams & Two-Dice Sums
Compound Probability Practice
Part A: Contingency & Venn Diagram Analysis
A survey of 100 high school seniors finds that 45 are enrolled in Finite Math (\(M\)), 30 participate in Academic Decathlon (\(D\)), and 15 do both.
\(M\) only
\(D\) only
15
Total Population \(n(S) = 100\)
1. How many students take Math only? How many do Decathlon only?
2. Calculate \(P(M \cup D)\), the probability a random senior does Math OR Decathlon:
3. Calculate \(P((M \cup D)')\), the probability a student does NEITHER:
Part B: The 36-Outcome Sample Space of Two Dice (Sum Distribution)
When rolling two standard 6-sided dice (Die 1 and Die 2), the sum ranges from 2 to 12 with \(6 \times 6 = 36\) ordered pairs. Complete the missing sums in the grid below:
| D1 \ D2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
a) \(P(\text{Sum} = 7)\):
b) \(P(\text{Sum} \ge 10)\):
c) \(P(\text{Sum is Even OR } > 9)\):
Part C: Finite Problem Set — Standard Deck
One card is drawn at random from a standard 52-card deck. Compute the following exact fraction probabilities. Show your addition rule setup.
1. \(P(\text{Face Card OR Red Card})\):
2. \(P(\text{Ace OR Spade})\):
Day 1 Mastery Check: Confident with Addition Rule Need clarification on overlap subtraction
Page 2 of 6
Finite Mathematics • Unit 3: Probability
The Multiplication Rule: Independent vs. Dependent
Day 2
Focus: Multi-Stage Experiments, Replacement vs. Non-Replacement
Materials: Deck of Cards • Calculator / RNG App
Independent Events (With Replacement) Sample Space Unchanged
Occurrence of \(A\) does NOT affect the probability of \(B\):
\(P(A \cap B) = P(A) \cdot P(B)\)
Dependent Events (Without Replacement) Sample Space Shrinks
Occurrence of \(A\) alters the probability of subsequent event \(B\):
\(P(A \cap B) = P(A) \cdot P(B|A)\)
1 Hands-On Lab: Two-Card Draw Comparison
Grab a standard 52-card deck with your partner. You will execute two sets of 6 trials to observe how replacing the first card changes compound outcomes.
Trial Set A: WITH Replacement
Draw Card 1 → Note suit → Return to deck & shuffle → Draw Card 2
| Trial | Card 1 | Card 2 | Both Red? (Y/N) |
|---|
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
Trial Set B: WITHOUT Replacement
Draw Card 1 → Note suit → DO NOT RETURN → Draw Card 2
| Trial | Card 1 | Card 2 | Both Red? (Y/N) |
|---|
| 1 | | | |
| 2 | | | |
| 3 | | | |
| 4 | | | |
| 5 | | | |
| 6 | | | |
1. Compute theoretical \(P(\text{Both Red})\) WITH replacement: Show fraction multiplication and decimal value:
2. Compute theoretical \(P(\text{Both Red})\) WITHOUT replacement: Explain what changed in numerator and denominator:
3. Compare the two theoretical probabilities. Which is smaller and why does this make intuitive sense?
Finite Mathematics • Student Investigation Packet Page 3 of 6
Day 2 Continued
Tree Diagram Models & Random Number Generator Simulations
Multi-Stage Compound Systems
Part A: The Urn Problem — Tree Diagram Construction
An urn contains 5 Green marbles and 3 Blue marbles (total 8). Two marbles are drawn sequentially without replacement. Complete the branch probabilities on the diagram and compute the probability of each path:
Draw 1: P(G) = 5/8 P(B) = 3/8
If G on Draw 1 → Draw 2: \(P(G|G) = \text{\_\_\_\_}\) Path G-G Probability: \(\text{\_\_\_\_}\)
If G on Draw 1 → Draw 2: \(P(B|G) = \text{\_\_\_\_}\) Path G-B Probability: \(\text{\_\_\_\_}\)
If B on Draw 1 → Draw 2: \(P(G|B) = \text{\_\_\_\_}\) Path B-G Probability: \(\text{\_\_\_\_}\)
If B on Draw 1 → Draw 2: \(P(B|B) = \text{\_\_\_\_}\) Path B-B Probability: \(\text{\_\_\_\_}\)
Path Rule Check
Multiply along branches; add paths that satisfy the event.
Sum of all 4 paths must equal exactly 1.0!
1. \(P(\text{Exactly one Green marble is drawn})\):
2. \(P(\text{At least one Blue marble is drawn})\) via complement:
2 RNG Simulation Lab: Multiple-Choice Guessing Streak
Command: randInt(1, 4)
Suppose a student completely guesses on a 3-question multiple-choice quiz (4 options each: A, B, C, D). Let the number 1 represent the correct answer. Using a calculator or online RNG, generate 3 random integers between 1 and 4 for 8 trials.
| Trial | Q1 Number | Q2 Number | Q3 Number | All 3 Correct? (All 1s) |
|---|
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| 8 | | | | |
a) Theoretical \(P(\text{All 3 correct})\) using multiplication rule:
b) Why does guessing on independent trials make streaks so rare?
Finite Mathematics • Student Investigation Packet Page 4 of 6
Finite Mathematics • Unit 3: Probability
Counting Principles: FCP, Permutations & Combinations
Day 3
Systematic Counting: Does Order Matter?
Tools: Scientific Calculator (\(nPr, nCr, !\))
Fundamental Counting (FCP)
If event 1 has \(n_1\) choices and event 2 has \(n_2\) choices:
\(N = n_1 \cdot n_2 \cdots n_k\)
Permutations (\(nPr\))
ORDER MATTERS! (Positions, ranks, sequences):
\(_n P_r = \frac{n!}{(n-r)!}\)
Combinations (\(nCr\))
ORDER DOES NOT MATTER! (Groups, hands):
\(_n C_r = \frac{n!}{r!(n-r)!}\)
1 Decision Lab: "Does Order Matter?" Sorting Matrix
For each scenario below, determine whether order matters. Label as Permutation (P) or Combination (C) and set up the mathematical notation.
| Real-World Scenario | Order Matters? | Type (P / C) | Formula Setup |
|---|
| Electing President, VP, and Treasurer from 12 candidates | | | |
| Selecting a 4-person cleanup committee from 15 volunteers | | | |
| Creating a 4-digit security PIN where digits cannot repeat | | | |
| Dealing a 5-card poker hand from a standard 52-card deck | | | |
2 RNG Discovery: Why Do We Divide by \(r!\)?
Consider selecting 3 distinct numbers from the set \(\{1, 2, 3, 4, 5\}\). Suppose you generate the numbers 2, 4, and 5.
Permutation View (Order Matters)
List all distinct orderings of \(\{2, 4, 5\}\):
Total orderings of 3 items \(= 3! = 3 \cdot 2 \cdot 1 = 6\)
Combination View (Order Irrelevant)
How many distinct groups/sets are represented?
All 6 permutations collapse into exactly 1 combination!
Explain in your own words why the combination formula \(_n C_r = \frac{_n P_r}{r!}\) divides by \(r!\):
Finite Mathematics • Student Investigation Packet Page 5 of 6
Day 3 Continued
Capstone: Computing Probability Using Counting Techniques
Synthesis & Mastery
The Finite Math Probability Principle
When outcomes in sample space \(S\) are equally likely, use combinations or permutations for both \(n(E)\) and \(n(S)\):
\(P(E) = \frac{\text{ways favorable event can occur}}{\text{total possible outcomes}} = \frac{n(E)}{n(S)}\)
Problem 1: Committee Representation
Combinations Approach
A high school student advisory board consists of 8 Seniors and 6 Juniors (14 total students). A subcommittee of 4 students is selected completely at random.
a) Total number of possible 4-person subcommittees in sample space \(n(S) = _{14}C_4\):
b) Number of ways to choose all 4 Seniors \(_{8}C_4\) and probability \(P(\text{All Seniors})\):
c) Number of ways to choose exactly 2 Seniors AND 2 Juniors: \((_8C_2 \cdot _6C_2)\) and resulting probability:
Problem 2: 5-Card Poker Hand Probability
Sample Space \(n(S) = _{52}C_5 = 2,598,960\)
Suppose 5 cards are dealt at random from a standard 52-card deck without replacement.
a) Probability that all 5 cards are Hearts (Flush): Hint: Choose 5 from the 13 hearts.
b) Probability of dealing all 4 Aces: Hint: Choose all 4 aces and 1 non-ace.
Three-Day Method Decider: When Do I Use Which Rule? Quick Reference
Addition Rule
Look for keyword "OR". Check for overlap to avoid double counting.
Multiplication Rule
Look for keyword "AND" across consecutive trials (check replacement!).
Permutations \(_nP_r\)
Count arrangements where order matters (titles, codes, finishes).
Combinations \(_nC_r\)
Count groupings where order does NOT matter (committees, hands).
Final Packet Self-Rating: • Ready for Unit Exam: 5 (Mastered) 4 3 (Reviewing)
Page 6 of 6 • Finite Mathematics
