Stats 1 Week 7: Probability Basics

0. Prerequisites

NOTE

What you need to know:

• Sets: Union ($\cup$), Intersection ($\cap$), Complement ($A^c$).
• Counting: Combinations (${}^n{C}_r$) are used heavily here.

Quick Refresher

• Sample Space ($S$): Set of all possible outcomes.
• Event ($E$): A subset of $S$.
• Probability $P(E)$: $\frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$.
• Axioms:
  1. $0\le P(E)\le 1$.
  2. $P(S)=1$.
  3. $P(A\cup B)=P(A)+P(B)$ (if Mutually Exclusive).

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1. Core Concepts

1.1 Types of Events

1. Mutually Exclusive (Disjoint): Cannot happen together. $A\cap B=\varnothing$.
   • Rule: $P(A\cup B)=P(A)+P(B)$.
2. Independent: Outcome of one doesn’t affect other.
   • Rule: $P(A\cap B)=P(A)\times P(B)$.
3. Complement: “Not A”.
   • Rule: $P(A^c)=1-P(A)$.

1.2 General Addition Rule

• Formula: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
• Logic: You counted the overlap twice, so subtract it once.

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2. Pattern Analysis & Goated Solutions

Pattern 1: “At Least One” (Complement Trick)

Context: “Probability of solving at least 1 problem out of 3?”

TIP

Mental Algorithm:

1. Identify Goal: “At least one” means 1, 2, or 3. Hard to calculate.
2. Invert: What is the opposite? “None”.
3. Calc Opposite: Find probability of failing ALL.
   • $P(\text{Fail All})=P(F_1)\times P(F_2)\times P(F_3)$ (assuming independence).
4. Subtract: $1-P(\text{None})$.

Example (Detailed Solution)

Problem: 3 students. Pass probs: 0.5, 0.6, 0.7. Prob at least one passes? Solution:

1. Fail Probs:
   • $1-0.5=0.5$.
   • $1-0.6=0.4$.
   • $1-0.7=0.3$.
2. Fail All: $0.5\times 0.4\times 0.3=0.06$.
3. At Least One: $1-0.06=0.94$. Answer: 0.94.

Pattern 2: Selection Probability (Hypergeometric)

Context: “Urn has 6 Red, 4 Black. Draw 3 balls. Prob of exactly 2 Red?”

TIP

Mental Algorithm:

1. Denominator (Total Ways): Select total needed from total available.
   • Total balls = 10. Need 3. ${\to }^{10}{C}_3$.
2. Numerator (Favorable Ways): Select specific colors.
   • Need 2 Red from 6 Red ${\to }^6{C}_2$.
   • Need 1 Black from 4 Black ${\to }^4{C}_1$.
3. Divide: $\frac{\text{Numerator}}{\text{Denominator}}$.

Example (Detailed Solution)

Problem: 6R, 4B. Draw 3. Prob 2R, 1B. Solution:

1. Total: ${}^{10}{C}_3=\frac{10\times 9\times 8}{6}=120$.
2. Favorable:
   • Red: ${}^6{C}_2=15$.
   • Black: ${}^4{C}_1=4$.
   • Total Fav: $15\times 4=60$.
3. Prob: $60/120=0.5$. Answer: 0.5.

Pattern 3: Union/Intersection Algebra

Context: “Given $P(A)=0.4,P(B)=0.5,P(A\cup B)=0.7$. Find $P(A\cap B)$.”

TIP

Mental Algorithm:

1. Write Formula: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
2. Plug & Solve:
   • $0.7=0.4+0.5-x$.
   • $0.7=0.9-x$.
   • $x=0.2$.

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3. Practice Exercises

1. Complement: Prob of rain is 0.3. Prob of NO rain?
   • Hint: $1-0.3=0.7$.
2. Independent: Coin flip (H) AND Die roll (6).
   • Hint: $0.5\times (1/6)=1/12$.
3. Mutually Exclusive: Draw Card. King OR Queen?
   • Hint: Can’t be both. $4/52+4/52=8/52$.

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🧠 Level Up: Advanced Practice

Question 1: The “At Least One” Probability

Problem: 3 shooters hit target with prob 0.4, 0.5, 0.6. Prob that target is hit at least once? Logic:

1. Complement: $P(\text{Hit})=1-P(\text{None Hit})$.
2. Miss Probs: $0.6,0.5,0.4$.
3. All Miss: $0.6\times 0.5\times 0.4=0.12$.
4. Result: $1-0.12=0.88$. Answer: 0.88.

Question 2: Axiomatic Check

Problem: $P(A)=0.6,P(B)=0.7$. Can they be mutually exclusive? Logic:

1. Sum: $P(A)+P(B)=1.3$.
2. Union Rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
3. Max Probability: $P(A\cup B)\le 1$.
4. Constraint: $1.3-P(A\cap B)\le 1⟹P(A\cap B)\ge 0.3$.
5. Conclusion: Intersection cannot be 0. They cannot be mutually exclusive. Answer: No.

Question 3: Odds vs Probability

Problem: Odds in favor of event A are 3:5. Find $P(A)$. Logic:

1. Odds: $n(A):n(A^{\prime })$.
2. Total: $3+5=8$.
3. Probability: $3/8$. Answer: $0.375$.