Probability Power Packet
Probability Blueprint
Regents Essential Reference
Ink-Friendly Edition
The Basics
General Prob. P(E) = favorable total
Not A (Comp.) P(A') = 1 − P(A)
Addition Rule (OR)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Use for the word "OR"
Conditional (GIVEN)
P(A|B) = P(A ∩ B) P(B)
Total is the "Given" group.
Independence Tests
Test 1: P(A ∩ B) = P(A) × P(B)
Test 2: P(A|B) = P(A)
Regents Exam Strategy
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1
Proof is Required: You MUST perform a calculation. Write out the numbers and show if they match.
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2
The "Given" Rule: To find P(A|B), circle Group B on the table. Its sum is your ONLY possible denominator.
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3
Significance: Results with probability < 0.05 are "unlikely" or "significant". Always compare your result to 0.05.
Algebra 2 Regents Prep Packet Page 1
Guided Note Lessons Name: __________________________________
1
The Basics and Complements
Example: A bag has 3 red, 5 blue, and 2 green marbles. Find the probability of NOT blue.
Work Space:
Notation Key
P(A') = 1 − P(A)
Total probability always sums to 1.
2
The Addition Rule (OR)
Example: 60% like Math, 50% like Science, 30% like both. Find P(Math OR Science).
Substitute and Solve:
Subtraction Logic
Subtraction prevents double-counting overlap.
Probability Power Packet Packet Page 2
3
Two-Way Frequency Tables
<table class="w-1/2 text-center border-collapse bg-white rounded-lg border border-slate-300 overflow-hidden text-xs font-bold shadow-md"><tbody><tr class="bg-slate-200 text-[10px] font-black uppercase border-b border-slate-300 text-slate-500 text-center"><td class="p-2 border-r border-slate-300">Group</td><td class="p-2 border-r border-slate-300">Bio</td><td class="p-2 border-r border-slate-300 text-slate-600 font-black">Chem</td><td class="p-2 text-slate-900">Total</td></tr><tr><td class="p-2 border-r border-slate-300 bg-slate-50 font-black text-slate-700">Jun.</td><td class="p-2 border-r border-slate-300 text-slate-800 text-center">40</td><td class="p-2 border-r border-slate-300 text-slate-800 text-center">10</td><td class="p-2 text-slate-800 text-center">50</td></tr><tr><td class="p-2 border-r border-slate-300 bg-slate-50 font-black text-slate-700">Sen.</td><td class="p-2 border-r border-slate-300 text-slate-800 text-center">15</td><td class="p-2 border-r border-slate-300 text-slate-800 text-center">35</td><td class="p-2 text-slate-800 text-center">50</td></tr><tr class="font-black border-t-2 border-slate-300 bg-slate-100 text-slate-900 text-[11px] text-center"><td>Total</td><td>55</td><td>45</td><td class="text-blue-900">100</td></tr></tbody></table>
Practice Intersection:
Find the probability a student is Senior AND in Chemistry.
4
Conditional Probability (GIVEN)
Example: Find P(Bio | Junior)
Rule of Thumb
"The group following the vertical bar becomes your ONLY world."
Algebra 2 Regents Prep Packet Page 3
5
Testing for Independence
Independence Practice:
Given P(A) = 0.6, P(B) = 0.5, and P(A ∩ B) = 0.3. Prove independence with calculation.
6
Making Decisions
The 5% Threshold
In Regents scenarios, if the result probability is less than 0.05, we say the result is "unlikely".
Written Justification Area:
Algebra 2 Regents Prep Packet Page 4
Regents Mastery Practice
Comprehensive Probability Review
Q1. A spinner has 12 equal sections (1 - 12). Find the probability of landing on a number greater than 8.
Q2. In a club, 40% play guitar, 30% play drums, and 10% play both. Find the probability a student plays Guitar OR Drums.
Q3. TWO-WAY TABLE CHALLENGE
<table class="w-1/2 text-center border-collapse bg-white rounded-xl border border-slate-300 text-[11px] font-black overflow-hidden shadow-sm"><tbody><tr class="bg-slate-200 border-b border-slate-300"><td></td><td>Sports</td><td>Music</td><td>Total</td></tr><tr class="border-b border-slate-100 font-bold text-slate-800"><td>Grade 9</td><td class="text-center">30</td><td class="text-center">20</td><td class="text-center">50</td></tr><tr class="border-b border-slate-100 font-bold text-slate-800"><td>Grade 10</td><td class="text-center">25</td><td class="text-center">25</td><td class="text-center">50</td></tr><tr class="bg-slate-100 border-t-2 border-slate-300 text-slate-900 font-black"><td>Total</td><td class="text-center">55</td><td class="text-center">45</td><td class="text-center">100</td></tr></tbody></table>
a) Find P(Music | Grade 10)
b) Find P(Grade 9 ∩ Sports)
Q4. Given P(A) = 0.6, P(B) = 0.5, and P(A ∩ B) = 0.3. Show work to prove if A and B are independent.
Algebra 2 Regents Prep Packet Page 5
Answer Key
FOR INSTRUCTOR REFERENCE
FINAL
Section 1: Foundations
Practice Q1:
4 12 = 1 3 ≈ 0.33
Guided Example #1:
1 − 5/10 = 0.50
Section 2: Addition
Practice Q2:
0.40 + 0.30 − 0.10 = 0.60
Guided #2 result: 0.80
Section 3: Tables
Practice a) 25 / 50 = 0.50
Practice b) 30 / 100 = 0.30
Guided #4: 40 / 50 = 0.80
Section 4: Independence
Q4 Logical Proof:
0.6 × 0.5 = 0.3
P(A ∩ B) is given as 0.3
Match: 0.3 = 0.3
Independent
Algebra 2 Regents Prep Packet Page 6: Answer Key
